3b1b-manim/old_projects/eoc/chapter4.py

from big_ol_pile_of_manim_imports import *

SINE_COLOR = BLUE
X_SQUARED_COLOR = GREEN
SUM_COLOR = YELLOW
PRODUCT_COLOR = YELLOW

class Chapter4OpeningQuote(OpeningQuote):
    CONFIG = {
        "quote" : [
            "Using the chain rule is like peeling an onion: ",
            "you have to deal with each layer at a time, and ",
            "if it is too big you will start crying."
        ],
        "author" : "(Anonymous professor)"
    }

class TransitionFromLastVideo(TeacherStudentsScene):
    def construct(self):
        simple_rules = VGroup(*list(map(TexMobject, [
            "\\frac{d(x^3)}{dx} = 3x^2",
            "\\frac{d(\\sin(x))}{dx} = \\cos(x)",
            "\\frac{d(1/x)}{dx} = -\\frac{1}{x^2}",
        ])))

        combination_rules = VGroup(*[
            TexMobject("\\frac{d}{dx}\\left(%s\\right)"%tex)
            for tex in [
                "\\sin(x) + x^2",
                "\\sin(x)(x^2)",
                "\\sin(x^2)",
            ]
        ])
        for rules in simple_rules, combination_rules:
            rules.arrange(buff = LARGE_BUFF)
            rules.next_to(self.get_teacher(), UP, buff = MED_LARGE_BUFF)
            rules.to_edge(LEFT)

        series = VideoSeries()
        series.to_edge(UP)
        last_video = series[2]
        last_video.save_state()
        this_video = series[3]
        brace = Brace(last_video)

        #Simple rules
        self.add(series)
        self.play(
            FadeIn(brace),
            last_video.set_color, YELLOW
        )
        for rule in simple_rules:
            self.play(
                Write(rule, run_time = 2),
                self.get_teacher().change_mode, "raise_right_hand",
                *[
                    ApplyMethod(pi.change_mode, "pondering")
                    for pi in self.get_students()
                ]
            )
            self.wait()
        self.wait(2)
        self.play(simple_rules.replace, last_video)
        self.play(
            last_video.restore,            
            Animation(simple_rules),
            brace.next_to, this_video, DOWN,
            this_video.set_color, YELLOW
        )

        #Combination rules
        self.play(
            Write(combination_rules),
            *[
                ApplyMethod(pi.change_mode, "confused")
                for pi in self.get_students()
            ]
        )
        self.wait(2)
        for rule in combination_rules:
            interior = VGroup(*rule[5:-1])
            added_anims = []
            if rule is combination_rules[-1]:
                inner_func = VGroup(*rule[-4:-2])
                self.play(inner_func.shift, 0.5*UP)
                added_anims = [
                    inner_func.shift, 0.5*DOWN,
                    inner_func.set_color, YELLOW
                ]
            self.play(
                interior.set_color, YELLOW,
                *added_anims,
                lag_ratio = 0.5
            )
            self.wait()
        self.wait()

        #Address subtraction and division
        subtraction = TexMobject("\\sin(x)", "-", "x^2")
        decomposed_subtraction = TexMobject("\\sin(x)", "+(-1)\\cdot", "x^2")
        pre_division = TexMobject("\\frac{\\sin(x)}{x^2}")
        division = VGroup(
            VGroup(*pre_division[:6]),
            VGroup(*pre_division[6:7]),
            VGroup(*pre_division[7:]),
        )
        pre_decomposed_division = TexMobject("\\sin(x)\\cdot\\frac{1}{x^2}")
        decomposed_division = VGroup(
            VGroup(*pre_decomposed_division[:6]),
            VGroup(*pre_decomposed_division[6:9]),
            VGroup(*pre_decomposed_division[9:]),
        )
        for mob in subtraction, decomposed_subtraction, division, decomposed_division:
            mob.next_to(
                VGroup(self.get_teacher(), self.get_students()[-1]),
                UP, buff = MED_LARGE_BUFF
            )

        top_group = VGroup(series, simple_rules, brace)
        combination_rules.save_state()
        self.play(
            top_group.next_to, FRAME_Y_RADIUS*UP, UP,
            combination_rules.to_edge, UP,
        )
        pairs = [
            (subtraction, decomposed_subtraction), 
            (division, decomposed_division)
        ]
        for question, answer in pairs:
            self.play(
                Write(question),
                combination_rules.fade, 0.2,
                self.get_students()[2].change_mode, "raise_right_hand",
                self.get_teacher().change_mode, "plain",
            )
            self.wait()
            answer[1].set_color(GREEN)
            self.play(
                Transform(question, answer),
                self.get_teacher().change_mode, "hooray",
                self.get_students()[2].change_mode, "plain",
            )
            self.wait()
            self.play(FadeOut(question))

        #Monstrous expression
        monster = TexMobject(
            "\\Big(",
            "e^{\\sin(x)} \\cdot",
            "\\cos\\Big(",
            "\\frac{1}{x^3}",
            " + x^3",
            "\\Big)",
            "\\Big)^4"
        )
        monster.next_to(self.get_pi_creatures(), UP)
        parts = [
            VGroup(*monster[3][2:]),
            VGroup(*monster[3][:2]),
            monster[4],
            VGroup(monster[2], monster[5]),
            monster[1],
            VGroup(monster[0], monster[6])
        ]
        modes = 3*["erm"] + 3*["pleading"]
        for part, mode in zip(parts, modes):
            self.play(
                FadeIn(part, lag_ratio = 0.5),
                self.get_teacher().change_mode, "raise_right_hand",
                *[
                    ApplyMethod(pi.change_mode, mode)
                    for pi in self.get_students()
                ]
            )
        self.wait()
        self.change_student_modes(*["happy"]*3)
        words = list(map(TextMobject, [
            "composition", "product", 
            "composition", "sum",
            "composition"
        ]))

        for word, part in zip(words, reversed(parts)):
            word.set_color(YELLOW)
            word.next_to(monster, UP)
            self.play(
                FadeIn(word),
                part.scale_in_place, 1.2,
                part.set_color, YELLOW
            )
            self.wait()
            self.play(*list(map(FadeOut, [word, part])))
        self.play(FadeOut(parts[0]))

        #Bring back combinations
        self.play(
            combination_rules.restore,
            *[
                ApplyMethod(pi_creature.change_mode, "pondering")
                for pi_creature in self.get_pi_creatures()
            ]
        )
        self.wait(2)

class DampenedSpring(Scene):
    def construct(self):
        compact_spring, extended_spring = [
            ParametricFunction(
                lambda t : (t/denom)*RIGHT+np.sin(t)*UP+np.cos(t)*OUT,
                t_max = 12*np.pi,
                color = GREY,
            ).shift(3*LEFT)
            for denom in (12.0, 2.0)
        ]
        for spring in compact_spring, extended_spring:
            spring.scale(0.5)
            spring.rotate(np.pi/6, UP)
            spring.set_color(GREY)
            spring.shift(-spring.points[0] + 3*LEFT)

        moving_spring = compact_spring.copy()

        def update_spring(spring, a):
            spring.interpolate(
                compact_spring, 
                extended_spring,
                0.5*(np.exp(-4*a)*np.cos(40*a)+1)
            )

        equation = TexMobject(
            "\\text{Length} = 2 + e^{-4t}\\cos(20t)"
        )
        equation.to_edge(UP)


        self.add(moving_spring, equation)
        self.play(UpdateFromAlphaFunc(
            moving_spring, update_spring, run_time = 10,
            rate_func=linear
        ))
        self.wait()

class ComingUp(Scene):
    def construct(self):
        rect = Rectangle(height = 9, width = 16)
        rect.set_stroke(WHITE)
        rect.set_height(FRAME_HEIGHT-2)
        title = TextMobject("Coming up...")
        title.to_edge(UP)
        rect.next_to(title, DOWN)

        self.play(Write(title))
        self.play(ShowCreation(rect))
        self.wait()

class PreSumRuleDiscussion(Scene):
    def construct(self):
        title = TextMobject("Sum rule")
        title.to_edge(UP)
        self.add(title)

        specific = TexMobject(
            "\\frac{d}{dx}(", "\\sin(x)", "+", "x^2", ")",
            "=", "\\cos(x)", "+", "2x"
        )
        general = TexMobject(
            "\\frac{d}{dx}(", "g(x)", "+", "h(x)", ")",
            "=", "\\frac{dg}{dx}", "+", "\\frac{dh}{dx}"
        )
        for formula in specific, general:
            formula[1].set_color(SINE_COLOR)
            formula[6].set_color(SINE_COLOR)
            formula[3].set_color(X_SQUARED_COLOR)
            formula[8].set_color(X_SQUARED_COLOR)
        VGroup(specific, general).arrange(DOWN, buff = LARGE_BUFF)

        #Add on rules
        self.add(specific)
        for i in 0, 4, 5:
            self.add(general[i])
        self.wait(2)
        for indices in [(1, 2, 3), (6,), (7, 8)]:
            self.play(*[
                ReplacementTransform(
                    specific[i].copy(), general[i]
                )
                for i in indices
            ])
            self.wait()

        #Highlight parts
        for i in 1, 3, -1, 6, 8:
            if i < 0:
                self.wait()
            else:
                part = specific[i]
                self.play(
                    part.set_color, YELLOW,
                    part.scale_in_place, 1.2,
                    rate_func = there_and_back
                )
        self.wait()

class SumRule(GraphScene):
    CONFIG = {
        "x_labeled_nums" : [],
        "y_labeled_nums" : [],
        "y_axis_label" : "",
        "x_max" : 4,
        "x_axis_width" : FRAME_WIDTH,
        "y_max" : 3,
        "graph_origin" : 2.5*DOWN + 2.5*LEFT,
        "graph_label_x_value" : 1.5,
        "example_input" : 0.5,
        "example_input_string" : "0.5",
        "dx" : 0.05,
        "v_lines_x_min" : -1,
        "v_lines_x_max" : 2,
        "graph_scale_factor" : 2,
        "tex_scale_factor" : 0.8,
    }
    def construct(self):
        self.write_function()
        self.add_graphs()
        self.zoom_in_on_graph()
        self.show_example_stacking()
        self.show_df()
        self.expand_derivative()

    def write_function(self):
        func_mob = TexMobject("f(x)", "=", "\\sin(x)", "+", "x^2")
        func_mob.scale(self.tex_scale_factor)
        func_mob.set_color_by_tex("f(x)", SUM_COLOR)
        func_mob.set_color_by_tex("\\sin(x)", SINE_COLOR)
        func_mob.set_color_by_tex("x^2", X_SQUARED_COLOR)
        func_mob.to_corner(UP+LEFT)
        self.add(func_mob)

        self.func_mob = func_mob

    def add_graphs(self):
        self.setup_axes()
        sine_graph = self.get_graph(np.sin, color = SINE_COLOR)
        parabola = self.get_graph(lambda x : x**2, color = X_SQUARED_COLOR)
        sum_graph = self.get_graph(
            lambda x : np.sin(x) + x**2, 
            color = SUM_COLOR
        )
        sine_label = self.get_graph_label(
            sine_graph, "\\sin(x)",
            x_val = self.graph_label_x_value,
            direction = UP+RIGHT,
            buff = 0,
        )
        sine_label.scale_in_place(self.tex_scale_factor)
        parabola_label = self.get_graph_label(
            parabola, "x^2", x_val = self.graph_label_x_value,
        )
        parabola_label.scale_in_place(self.tex_scale_factor)

        graphs = VGroup(sine_graph, parabola)
        labels = VGroup(sine_label, parabola_label)
        for label in labels:
            label.add_background_rectangle()

        for graph, label in zip(graphs, labels):
            self.play(
                ShowCreation(graph),
                Write(label)
            )
        self.wait()

        num_lines = (self.v_lines_x_max-self.v_lines_x_min)/self.dx
        sine_v_lines, parabox_v_lines = v_line_sets = [
            self.get_vertical_lines_to_graph(
                graph, 
                x_min = self.v_lines_x_min,
                x_max = self.v_lines_x_max, 
                num_lines = num_lines,
                stroke_width = 2
            )
            for graph in graphs
        ]
        sine_v_lines.shift(0.02*RIGHT)
        for v_lines in v_line_sets:
            self.play(ShowCreation(v_lines), Animation(labels))
        self.wait()
        self.play(*it.chain(
            [
                ApplyMethod(l2.move_to, l1.get_top(), DOWN)
                for l1, l2, in zip(*v_line_sets)
            ],
            [graph.fade for graph in graphs],
            [Animation(labels)]
        ))
        self.wait()

        self.play(ShowCreation(sum_graph))
        self.wait()

        self.sum_graph = sum_graph
        self.parabola = parabola
        self.sine_graph = sine_graph
        self.graph_labels = labels
        self.v_line_sets = v_line_sets

    def zoom_in_on_graph(self):
        graph_parts = VGroup(
            self.axes, 
            self.sine_graph, self.parabola, self.sum_graph,
            *self.v_line_sets
        )
        graph_parts.remove(self.func_mob, *self.graph_labels)
        graph_parts.generate_target()
        self.graph_labels.generate_target()
        for mob in graph_parts, self.graph_labels:
            mob.target.scale(
                self.graph_scale_factor, 
                about_point = self.graph_origin,
            )
        for mob in self.graph_labels.target:
            mob.scale(
                1./self.graph_scale_factor,
                about_point = mob.get_bottom()
            )
            mob.shift_onto_screen()
        self.play(*list(map(MoveToTarget, [
            graph_parts, self.graph_labels
        ])))
        self.wait()

    def show_example_stacking(self):
        v_line_sets = self.v_line_sets
        num_lines = len(v_line_sets[0])
        example_v_lines, nudged_v_lines = [
            VGroup(*[v_lines[index] for v_lines in v_line_sets])
            for index in (num_lines/2, num_lines/2+1)
        ]
        for line in nudged_v_lines:
            line.save_state()
        sine_lines, parabola_lines = [
            VGroup(example_v_lines[i], nudged_v_lines[i])
            for i in (0, 1)
        ]
        faders = VGroup(*[line for line in it.chain(*v_line_sets) if line not in example_v_lines])
        label_groups = []
        for line, tex, vect in zip(sine_lines, ["", "+dx"], [LEFT, RIGHT]):
            dot = Dot(line.get_bottom(), radius = 0.03, color = YELLOW)
            label = TexMobject(
                "x=" + str(self.example_input) + tex
            )
            label.next_to(dot, DOWN+vect, buff = MED_LARGE_BUFF)
            arrow = Arrow(
                label.get_corner(UP-vect), dot,
                buff = SMALL_BUFF,
                color = WHITE,
                tip_length = 0.1
            )
            label_groups.append(VGroup(label, arrow, dot))

        line_tex_direction_triplets = [
            (sine_lines[0], "\\sin(0.5)", LEFT),
            (sine_lines[1], "\\sin(0.5+dx)", RIGHT),
            (parabola_lines[0], "(0.5)^2", LEFT),
            (parabola_lines[1], "(0.5+dx)^2", RIGHT),
        ]
        for line, tex, direction in line_tex_direction_triplets:
            line.brace = Brace(
                line, direction,
                buff = SMALL_BUFF,
                min_num_quads = 2,
            )
            line.brace.set_color(line.get_color())
            line.brace.add_background_rectangle()
            line.brace_text = line.brace.get_text("$%s$"%tex)
            line.brace_text.scale(
                self.tex_scale_factor,
                about_point = line.brace_text.get_edge_center(-direction)
            )
            line.brace_text.add_background_rectangle()
            line.brace_anim = MaintainPositionRelativeTo(
                VGroup(line.brace, line.brace_text), line
            )

        ##Look at example lines
        self.play(
            example_v_lines.set_stroke, None, 4,
            faders.fade,
            Animation(self.graph_labels),
            Write(label_groups[0]),
        )
        for line in example_v_lines:
            line.save_state()
        self.wait()
        self.play(
            GrowFromCenter(sine_lines[0].brace),
            Write(sine_lines[0].brace_text),
        )
        self.wait()
        self.play(
            sine_lines[0].shift, UP+4*LEFT,
            sine_lines[0].brace_anim,
            parabola_lines[0].move_to, sine_lines[0], DOWN
        )
        self.wait()
        parabola_lines[0].brace_anim.update(1)
        self.play(
            GrowFromCenter(parabola_lines[0].brace),
            Write(parabola_lines[0].brace_text),
        )
        self.wait()
        self.play(*it.chain(*[
            [line.restore, line.brace_anim]
            for line in example_v_lines
        ]))

        ## Nudged_lines
        self.play(
            Write(label_groups[1]),
            *it.chain(*[
                [line.restore, line.set_stroke, None, 4]
                for line in nudged_v_lines
            ])
        )
        self.wait()
        for line in nudged_v_lines:
            self.play(
                GrowFromCenter(line.brace),
                Write(line.brace_text)
            )
        self.wait()

        self.sine_lines = sine_lines
        self.parabola_lines = parabola_lines

    def show_df(self):
        sine_lines = self.sine_lines
        parabola_lines = self.parabola_lines

        df, equals, d_sine, plus, d_x_squared = deriv_mob = TexMobject(
            "df", "=", "d(\\sin(x))", "+", "d(x^2)"
        )
        df.set_color(SUM_COLOR)
        d_sine.set_color(SINE_COLOR)
        d_x_squared.set_color(X_SQUARED_COLOR)
        deriv_mob.scale(self.tex_scale_factor)
        deriv_mob.next_to(
            self.func_mob, DOWN, 
            buff = MED_LARGE_BUFF, 
            aligned_edge = LEFT
        )
        for submob in deriv_mob:
            submob.add_to_back(BackgroundRectangle(submob))

        df_lines = self.show_difference(parabola_lines, df, equals)
        self.wait()
        self.play(FadeOut(df_lines))
        self.play(
            parabola_lines[0].shift,
            (parabola_lines[1].get_bottom()[1]-parabola_lines[0].get_bottom()[1])*UP,
            parabola_lines[0].brace_anim
        )
        d_sine_lines = self.show_difference(sine_lines, d_sine, plus)
        d_x_squared_lines = self.show_difference(parabola_lines, d_x_squared, VGroup())
        self.wait()

        self.deriv_mob = deriv_mob
        self.d_sine_lines = d_sine_lines
        self.d_x_squared_lines = d_x_squared_lines

    def show_difference(self, v_lines, target_tex, added_tex):
        distance = v_lines[1].get_top()[1]-v_lines[0].get_top()[1]
        h_lines = VGroup(*[
            DashedLine(ORIGIN, 2*RIGHT, stroke_width = 3)
            for x in range(2)
        ])
        h_lines.arrange(DOWN, buff = distance)
        h_lines.move_to(v_lines[1].get_top(), UP+RIGHT)

        brace = Brace(h_lines, LEFT)
        brace_text = target_tex.copy()
        brace_text.next_to(brace, LEFT)

        self.play(ShowCreation(h_lines))
        self.play(GrowFromCenter(brace), Write(brace_text))
        self.wait()
        self.play(
            ReplacementTransform(brace_text.copy(), target_tex),
            Write(added_tex)
        )
        return VGroup(h_lines, brace, brace_text)

    def expand_derivative(self):
        expanded_deriv = TexMobject(
            "df", "=", "\\cos(x)", "\\,dx", "+", "2x", "\\,dx"
        )
        expanded_deriv.set_color_by_tex("df", SUM_COLOR)
        VGroup(*expanded_deriv[2:4]).set_color(SINE_COLOR)
        VGroup(*expanded_deriv[5:7]).set_color(X_SQUARED_COLOR)
        expanded_deriv.scale(self.tex_scale_factor)
        expanded_deriv.next_to(
            self.deriv_mob, DOWN,
            buff = MED_LARGE_BUFF,
            aligned_edge = LEFT
        )
        background_rect = BackgroundRectangle(expanded_deriv)

        rearranged_deriv = TexMobject(
            "{df \\over", "dx}", "=", "\\cos(x)", "+", "2x"
        )
        rearranged_deriv[0].set_color(SUM_COLOR)
        rearranged_deriv[3].set_color(SINE_COLOR)
        rearranged_deriv[5].set_color(X_SQUARED_COLOR)
        rearranged_deriv.scale(self.tex_scale_factor)
        rearranged_deriv.move_to(expanded_deriv, UP+LEFT)
        deriv_target_indices = [0, 2, 3, 1, 4, 5, 1]

        self.play(
            FadeIn(
                background_rect, 
                rate_func = squish_rate_func(smooth, 0.6, 1)
            ),
            Write(expanded_deriv)
        )
        self.wait()

        tex_group_pairs = [
            ("\\cos(0.5)dx", self.d_sine_lines),
            ("2(0.5)dx", self.d_x_squared_lines),
        ]
        def indicate(mob):
            self.play(
                mob.set_color, YELLOW,
                mob.scale_in_place, 1.2,
                rate_func = there_and_back
            )
        for tex, group in tex_group_pairs:
            old_label = group[-1]
            new_label = TexMobject(tex)
            pre_dx = VGroup(*new_label[:-2])
            dx = VGroup(*new_label[-2:])
            new_label.add_background_rectangle()
            new_label.scale(self.tex_scale_factor)
            new_label.move_to(old_label, RIGHT)
            new_label.set_color(old_label.get_color())

            self.play(FocusOn(old_label))
            indicate(old_label)
            self.wait()
            self.play(FadeOut(old_label))
            self.play(FadeIn(new_label))
            self.wait()
            indicate(dx)
            self.wait()
            indicate(pre_dx)
            self.wait()
        self.wait()
        self.play(*[
            Transform(
                expanded_deriv[i], rearranged_deriv[j],
                path_arc = -np.pi/2
            )
            for i, j in enumerate(deriv_target_indices)
        ])
        self.wait()

class DiscussProducts(TeacherStudentsScene):
    def construct(self):
        wrong_product_rule = TexMobject(
            "\\frac{d(\\sin(x)x^2)}{dx}", 
            "\\ne",
            "\\left(\\frac{d(\\sin(x))}{dx}\\right)",
            "\\left(\\frac{d(x^2)}{dx}\\right)",
        )
        not_equals = wrong_product_rule[1]
        wrong_product_rule[2].set_color(SINE_COLOR)
        wrong_product_rule[3].set_color(X_SQUARED_COLOR)
        wrong_product_rule.next_to(
            self.get_teacher().get_corner(UP+LEFT),
            UP,
            buff = MED_LARGE_BUFF
        ).shift_onto_screen()

        self.teacher_says(
            "Products are a bit different",
            target_mode = "sassy"
        )
        self.wait(2)
        self.play(RemovePiCreatureBubble(
            self.get_teacher(), 
            target_mode = "raise_right_hand"
        ))
        self.play(Write(wrong_product_rule))
        self.change_student_modes(
            "pondering", "confused", "erm",
            added_anims = [
                not_equals.scale_in_place, 1.3,
                not_equals.set_color, RED
            ]
        )
        self.wait()
        self.teacher_says(
            "Think about the \\\\ underlying meaning",
            bubble_kwargs = {"height" : 3},
            added_anims = [
                wrong_product_rule.scale, 0.7,
                wrong_product_rule.to_corner, UP+LEFT
            ]
        )
        self.change_student_modes(*["pondering"]*3)
        self.wait(2)

class NotGraphsForProducts(GraphScene):
    CONFIG = {
        "y_max" : 25,
        "x_max" : 7,
    }
    def construct(self):
        self.setup_axes()
        sine_graph = self.get_graph(np.sin, color = SINE_COLOR)
        sine_graph.label = self.get_graph_label(
            sine_graph, "\\sin(x)",
            x_val = 3*np.pi/2,
            direction = DOWN
        )
        parabola = self.get_graph(
            lambda x : x**2, color = X_SQUARED_COLOR
        )
        parabola.label = self.get_graph_label(
            parabola, "x^2",
            x_val = 2.5,
            direction = UP+LEFT,
        )
        product_graph = self.get_graph(
            lambda x : np.sin(x)*(x**2), color = PRODUCT_COLOR
        )
        product_graph.label = self.get_graph_label(
            product_graph, "\\sin(x)x^2",
            x_val = 2.8,
            direction = UP+RIGHT,
            buff = 0
        )

        graphs = [sine_graph, parabola, product_graph]
        for graph in graphs:
            self.play(
                ShowCreation(graph),
                Write(graph.label, run_time = 2)
            )
        self.wait()

        everything = VGroup(*[m for m in self.get_mobjects() if not m.is_subpath])
        words = TextMobject("Not the best visualization")
        words.scale(1.5)
        words.shift(FRAME_Y_RADIUS*UP/2)
        words.add_background_rectangle()
        words.set_color(RED)
        self.play(
            everything.fade,
            Write(words)
        )
        self.wait()

class ConfusedMorty(Scene):
    def construct(self):
        morty = Mortimer()
        self.add(morty)
        self.wait()
        self.play(morty.change_mode, "confused")
        self.play(Blink(morty))
        self.wait(2)

class IntroduceProductAsArea(ReconfigurableScene):
    CONFIG = {
        "top_func" : np.sin,
        "top_func_label" : "\\sin(x)",
        "top_func_nudge_label" : "d(\\sin(x))",
        "top_func_derivative" : "\\cos(x)",
        "side_func" : lambda x : x**2,
        "side_func_label" : "x^2",
        "side_func_nudge_label" : "d(x^2)",
        "side_func_derivative" : "2x",
        "x_unit_to_space_unit" : 3,
        "box_kwargs" : {
            "fill_color" : YELLOW,
            "fill_opacity" : 0.75,
            "stroke_width" : 1,
        },
        "df_box_kwargs" : {
            "fill_color" : GREEN,
            "fill_opacity" : 0.75,
            "stroke_width" : 0,
        },
        "box_corner_location" : 6*LEFT+2.5*UP,
        "slider_center" : 3.5*RIGHT+2*DOWN,
        "slider_width" : 6,
        "slider_x_max" : 3,
        "x_slider_handle_height" : 0.25,
        "slider_handle_color" : BLUE,
        "default_x" : .75,
        "dx" : 0.1,
        "tiny_dx" : 0.01,
    }
    def construct(self):
        self.introduce_box()
        self.talk_though_sine()
        self.define_f_of_x()
        self.nudge_x()
        self.write_df()
        self.show_thinner_dx()
        self.expand_derivative()
        self.write_derivative_abstractly()
        self.write_mneumonic()

    def introduce_box(self):
        box, labels = self.box_label_group = self.get_box_label_group(self.default_x)
        self.x_slider = self.get_x_slider(self.default_x)

        self.play(Write(labels))
        self.play(DrawBorderThenFill(box))
        self.wait()
        for mob in self.x_slider:
            self.play(Write(mob, run_time = 1))
        self.wait()
        for new_x in 0.5, 2, self.default_x:
            self.animate_x_change(
                new_x, run_time = 2
            )
        self.wait()

    def talk_though_sine(self):
        x_axis = self.x_slider[0]
        graph = FunctionGraph(
            np.sin, x_min = 0, x_max = np.pi,
            color = SINE_COLOR
        )
        scale_factor = self.x_slider.get_width()/self.slider_x_max
        graph.scale(scale_factor)
        graph.move_to(x_axis.number_to_point(0), DOWN+LEFT)

        label = TexMobject("\\sin(x)")
        label.set_color(SINE_COLOR)
        label.next_to(graph, UP)

        y_axis = x_axis.copy()
        y_axis.remove(*y_axis.numbers)

        v_line = Line(ORIGIN, UP, color = WHITE, stroke_width = 2)
        def v_line_update(v_line):
            x = x_axis.point_to_number(self.x_slider[1].get_top())
            v_line.set_height(np.sin(x)*scale_factor)
            v_line.move_to(x_axis.number_to_point(x), DOWN)
        v_line_update(v_line)

        self.play(
            Rotate(y_axis, np.pi/2, about_point = y_axis.get_left()),
            Animation(x_axis)
        )
        self.play(
            ShowCreation(graph),
            Write(label, run_time = 1)
        )
        self.play(ShowCreation(v_line))
        for x, rt in zip([0.25, np.pi/2, 3, self.default_x], [2, 4, 4, 2]):
            self.animate_x_change(
                x, run_time = rt, 
                added_anims = [
                    UpdateFromFunc(v_line, v_line_update)
                ]
            )
            self.wait()
        self.play(*it.chain(
            list(map(FadeOut, [y_axis, graph, label, v_line])),
            [Animation(x_axis)]
        ))
        self.wait()
        for x in 1, 0.5, self.default_x:
            self.animate_x_change(x)
        self.wait()

    def define_f_of_x(self):
        f_def = TexMobject(
            "f(x)", "=",
            self.top_func_label,
            self.side_func_label,
            "=", 
            "\\text{Area}"
        )
        f_def.to_corner(UP+RIGHT)
        f_def[-1].set_color(self.box_kwargs["fill_color"])

        box, labels = self.box_label_group

        self.play(Write(VGroup(*f_def[:-1])))
        self.play(Transform(
            box.copy().set_fill(opacity = 0), f_def[-1],
            run_time = 1.5,
        ))
        self.wait()

        self.f_def = f_def

    def nudge_x(self):
        box, labels = self.box_label_group
        nudge_label_group = self.get_nudge_label_group()
        original_dx = self.dx
        self.dx = self.tiny_dx
        thin_df_boxes = self.get_df_boxes()
        self.dx = original_dx
        df_boxes = self.get_df_boxes()
        right_box, corner_box, right_box = df_boxes

        self.play(FocusOn(nudge_label_group))
        self.play(*list(map(GrowFromCenter, nudge_label_group)))
        self.animate_x_change(
            self.default_x+self.dx,
            rate_func = there_and_back,
            run_time = 2,
            added_anims = [Animation(nudge_label_group)]
        ) 
        self.wait()
        self.play(
            ReplacementTransform(thin_df_boxes, df_boxes),
            VGroup(*labels[1]).shift, right_box.get_width()*RIGHT,
        )
        self.play(
            df_boxes.space_out_submobjects, 1.1,
            df_boxes.move_to, box, UP+LEFT,
        )
        self.wait()

        self.df_boxes = df_boxes
        self.df_box_labels = self.get_df_box_labels(df_boxes)
        self.x_slider.add(nudge_label_group)

    def get_nudge_label_group(self):
        line, triangle, x_mob = self.x_slider
        dx_line = Line(*[
            line.number_to_point(self.x_slider.x_val + num)
            for num in (0, self.dx,)
        ])
        dx_line.set_stroke(
            self.df_box_kwargs["fill_color"], 
            width = 6
        )
        brace = Brace(dx_line, UP, buff = SMALL_BUFF)
        brace.stretch_to_fit_height(0.2)
        brace.next_to(dx_line, UP, buff = SMALL_BUFF)
        brace.set_stroke(width = 1)
        dx = TexMobject("dx")
        dx.scale(0.7)
        dx.next_to(brace, UP, buff = SMALL_BUFF)
        dx.set_color(dx_line.get_color())

        return VGroup(dx_line, brace, dx)

    def get_df_boxes(self):
        box, labels = self.box_label_group
        alt_box = self.get_box(self.x_slider.x_val + self.dx)

        h, w = box.get_height(), box.get_width()
        dh, dw = alt_box.get_height()-h, alt_box.get_width()-w
        heights_and_widths = [(dh, w), (dh, dw), (h, dw)]
        vects = [DOWN, DOWN+RIGHT, RIGHT]
        df_boxes = VGroup(*[
            Rectangle(
                height = height, width = width, **self.df_box_kwargs
            ).next_to(box, vect, buff = 0)
            for (height, width), vect in zip(
                heights_and_widths, vects
            )
        ])
        return df_boxes

    def get_df_box_labels(self, df_boxes):
        bottom_box, corner_box, right_box = df_boxes
        result = VGroup()
        quads = [
            (right_box, UP, self.top_func_nudge_label, LEFT),
            (corner_box, RIGHT, self.side_func_nudge_label, ORIGIN),
        ]
        for box, vect, label_tex, aligned_edge in quads:
            brace = Brace(box, vect)
            label = TexMobject(label_tex)
            label.next_to(
                brace, vect,
                aligned_edge = aligned_edge,
                buff = SMALL_BUFF
            )
            label.set_color(df_boxes[0].get_color())
            result.add(VGroup(brace, label))
        return result

    def write_df(self):
        deriv = TexMobject(
            "df", "=", 
            self.top_func_label, 
            self.side_func_nudge_label,
            "+",
            self.side_func_label,
            self.top_func_nudge_label,
        )
        deriv.scale(0.9)
        deriv.next_to(self.f_def, DOWN, buff = LARGE_BUFF)
        deriv.to_edge(RIGHT)
        for submob, tex in zip(deriv, deriv.expression_parts):
            if tex.startswith("d"):
                submob.set_color(self.df_box_kwargs["fill_color"])
        bottom_box_area = VGroup(*deriv[2:4])
        right_box_area = VGroup(*deriv[5:7])

        bottom_box, corner_box, right_box = self.df_boxes
        plus = TexMobject("+").set_fill(opacity = 0)
        df_boxes_copy = VGroup(
            bottom_box.copy(),
            plus,
            right_box.copy(),
            plus.copy(),
            corner_box.copy(),
        )

        self.deriv = deriv
        self.df_boxes_copy = df_boxes_copy
        box, labels = self.box_label_group
        self.full_box_parts = VGroup(*it.chain(
            [box], self.df_boxes, labels, self.df_box_labels
        ))

        self.play(Write(VGroup(*deriv[:2])))
        self.play(
            df_boxes_copy.arrange,
            df_boxes_copy.set_fill, None, self.df_box_kwargs["fill_opacity"],
            df_boxes_copy.next_to, deriv[1]
        )
        deriv.submobjects[4] = df_boxes_copy[1]
        self.wait()

        self.set_color_right_boxes()
        self.set_color_bottom_boxes()
        self.describe_bottom_box(bottom_box_area)
        self.describe_right_box(right_box_area)
        self.ignore_corner()

        # self.add(deriv)

    def set_color_boxes_and_label(self, boxes, label):
        boxes.save_state()
        label.save_state()

        self.play(GrowFromCenter(label))
        self.play(
            boxes.set_color, RED,
            label.set_color, RED,
        )
        self.play(
            label[1].scale_in_place, 1.1,
            rate_func = there_and_back
        )
        self.play(boxes.restore, label.restore)
        self.wait()

    def set_color_right_boxes(self):
        self.set_color_boxes_and_label(
            VGroup(*self.df_boxes[1:]),
            self.df_box_labels[0]
        )

    def set_color_bottom_boxes(self):
        self.set_color_boxes_and_label(
            VGroup(*self.df_boxes[:-1]),
            self.df_box_labels[1]
        )

    def describe_bottom_box(self, bottom_box_area):
        bottom_box = self.df_boxes[0]
        bottom_box_copy = self.df_boxes_copy[0]
        other_box_copies = VGroup(*self.df_boxes_copy[1:])
        top_label = self.box_label_group[1][0]
        right_label = self.df_box_labels[1]

        faders = VGroup(*[m for m in self.full_box_parts if m not in [bottom_box, top_label, right_label]])
        faders.save_state()

        self.play(faders.fade, 0.8)
        self.wait()
        self.play(FocusOn(bottom_box_copy))
        self.play(
            ReplacementTransform(bottom_box_copy, bottom_box_area),
            other_box_copies.next_to, bottom_box_area, RIGHT
        )
        self.wait()
        self.play(faders.restore)

    def describe_right_box(self, right_box_area):
        right_box = self.df_boxes[2]
        right_box_copy = self.df_boxes_copy[2]
        right_box_area.next_to(self.df_boxes_copy[1])
        other_box_copies = VGroup(*self.df_boxes_copy[3:])
        top_label = self.df_box_labels[0]
        right_label = self.box_label_group[1][1]

        faders = VGroup(*[m for m in self.full_box_parts if m not in [right_box, top_label, right_label]])
        faders.save_state()

        self.play(faders.fade, 0.8)
        self.wait()
        self.play(FocusOn(right_box_copy))
        self.play(
            ReplacementTransform(right_box_copy, right_box_area),                        
            other_box_copies.next_to, right_box_area, DOWN, 
            MED_SMALL_BUFF, RIGHT,

        )
        self.wait()
        self.play(faders.restore)

    def ignore_corner(self):
        corner = self.df_boxes[1]
        corner.save_state()
        corner_copy = VGroup(*self.df_boxes_copy[-2:])
        words = TextMobject("Ignore")
        words.set_color(RED)
        words.next_to(corner_copy, LEFT, buff = LARGE_BUFF)
        words.shift(MED_SMALL_BUFF*DOWN)
        arrow = Arrow(words, corner_copy, buff = SMALL_BUFF, color = RED)

        self.play(
            corner.set_color, RED,
            corner_copy.set_color, RED,
        )
        self.wait()
        self.play(Write(words), ShowCreation(arrow))
        self.wait()
        self.play(*list(map(FadeOut, [words, arrow, corner_copy])))
        self.wait()
        corner_copy.set_color(BLACK)

    def show_thinner_dx(self):
        self.transition_to_alt_config(dx = self.tiny_dx)

    def expand_derivative(self):
        # self.play(
        #     self.deriv.next_to, self.f_def, DOWN, MED_LARGE_BUFF,
        #     self.deriv.shift_onto_screen
        # )
        # self.wait()

        expanded_deriv = TexMobject(
            "df", "=", 
            self.top_func_label, 
            self.side_func_derivative,
            "\\,dx",
            "+", 
            self.side_func_label,
            self.top_func_derivative, 
            "\\,dx"
        )
        final_deriv = TexMobject(
            "{df \\over ", "dx}", "=", 
            self.top_func_label, 
            self.side_func_derivative,
            "+", 
            self.side_func_label,
            self.top_func_derivative, 
        )
        color = self.deriv[0].get_color()
        for new_deriv in expanded_deriv, final_deriv:        
            for submob, tex in zip(new_deriv, new_deriv.expression_parts):
                for substr in "df", "dx", self.top_func_derivative, self.side_func_derivative:
                    if substr in tex:
                        submob.set_color(color)
            new_deriv.scale(0.9)
            new_deriv.next_to(self.deriv, DOWN, buff = MED_LARGE_BUFF)
            new_deriv.shift_onto_screen()

        def indicate(mob):
            self.play(
                mob.scale_in_place, 1.2,
                mob.set_color, YELLOW,
                rate_func = there_and_back
            )


        for index in 6, 3:
            self.deriv.submobjects.insert(
                index+1, self.deriv[index].copy()
            )
        non_deriv_indices = list(range(len(expanded_deriv)))
        for indices in [(3, 4), (7, 8)]:
            top_part = VGroup()
            bottom_part = VGroup()            
            for i in indices:
                non_deriv_indices.remove(i)
                top_part.add(self.deriv[i].copy())
                bottom_part.add(expanded_deriv[i])
            self.play(top_part.move_to, bottom_part)
            self.wait()
            indicate(top_part)
            self.wait()
            self.play(ReplacementTransform(top_part, bottom_part))
            self.wait()
        top_part = VGroup()
        bottom_part = VGroup()
        for i in non_deriv_indices:
            top_part.add(self.deriv[i].copy())
            bottom_part.add(expanded_deriv[i])
        self.play(ReplacementTransform(
            top_part, bottom_part
        ))

        self.wait()
        self.play(*[
            ReplacementTransform(
                expanded_deriv[i], final_deriv[j],
                path_arc = -np.pi/2
            )
            for i, j in [
                (0, 0),
                (1, 2),
                (2, 3),
                (3, 4),
                (4, 1),
                (5, 5),
                (6, 6),
                (7, 7),
                (8, 1),
            ]
        ])
        self.wait()
        for index in 0, 1, 3, 4, 6, 7:
            indicate(final_deriv[index])
        self.wait()

    def write_derivative_abstractly(self):
        self.transition_to_alt_config(
            return_to_original_configuration = False,
            top_func_label = "g(x)",
            top_func_nudge_label = "dg",
            top_func_derivative = "\\frac{dg}{dx}",
            side_func_label = "h(x)",
            side_func_nudge_label = "dh",
            side_func_derivative = "\\frac{dh}{dx}",
        )
        self.wait()

    def write_mneumonic(self):
        morty = Mortimer()
        morty.scale(0.7)
        morty.to_edge(DOWN)
        morty.shift(2*LEFT)
        words = TextMobject(
            "``Left ", "d(Right) ", "+", " Right ", "d(Left)", "''",
            arg_separator = ""
        )
        VGroup(words[1], words[4]).set_color(self.df_boxes[0].get_color())
        words.scale(0.7)
        words.next_to(morty.get_corner(UP+LEFT), UP)
        words.shift_onto_screen()

        self.play(FadeIn(morty))
        self.play(
            morty.change_mode, "raise_right_hand",
            Write(words)
        )
        self.wait()

    ###############

    def animate_x_change(
        self, target_x, 
        box_label_group = None, 
        x_slider = None,
        **kwargs
        ):
        box_label_group = box_label_group or self.box_label_group
        x_slider = x_slider or self.x_slider
        kwargs["run_time"] = kwargs.get("run_time", 2)
        added_anims = kwargs.get("added_anims", [])

        start_x = x_slider.x_val
        def update_box_label_group(box_label_group, alpha):
            new_x = interpolate(start_x, target_x, alpha)
            new_box_label_group = self.get_box_label_group(new_x)
            Transform(box_label_group, new_box_label_group).update(1)
        
        def update_x_slider(x_slider, alpha):
            new_x = interpolate(start_x, target_x, alpha)
            new_x_slider = self.get_x_slider(new_x)
            Transform(x_slider, new_x_slider).update(1)

        self.play(
            UpdateFromAlphaFunc(
                box_label_group, 
                update_box_label_group
            ),
            UpdateFromAlphaFunc(
                x_slider, 
                update_x_slider
            ),
            *added_anims,
            **kwargs
        )
        x_slider.x_val = target_x

    def get_x_slider(self, x):
        numbers = list(range(int(self.slider_x_max) + 1))
        line = NumberLine(
            x_min = 0,
            x_max = self.slider_x_max,
            unit_size = float(self.slider_width)/self.slider_x_max,
            color = GREY,
            numbers_with_elongated_ticks = numbers,
            tick_frequency = 0.25,
        )
        line.add_numbers(*numbers)
        line.numbers.next_to(line, UP, buff = SMALL_BUFF)
        for number in line.numbers:
            number.add_background_rectangle()
        line.move_to(self.slider_center)

        triangle = RegularPolygon(
            3, start_angle = np.pi/2,
            fill_color = self.slider_handle_color,
            fill_opacity = 0.8,
            stroke_width = 0,
        )
        triangle.set_height(self.x_slider_handle_height)
        triangle.move_to(line.number_to_point(x), UP)

        x_mob = TexMobject("x")
        x_mob.next_to(triangle, DOWN, buff = SMALL_BUFF)

        result = VGroup(line, triangle, x_mob)
        result.x_val = x
        return result

    def get_box_label_group(self, x):
        box = self.get_box(x)
        labels = self.get_box_labels(box)
        return VGroup(box, labels)

    def get_box(self, x):
        box = Rectangle(
            width = self.x_unit_to_space_unit * self.top_func(x),
            height = self.x_unit_to_space_unit * self.side_func(x),
            **self.box_kwargs
        )
        box.move_to(self.box_corner_location, UP+LEFT)
        return box

    def get_box_labels(self, box):
        result = VGroup()
        for label_tex, vect in (self.top_func_label, UP), (self.side_func_label, RIGHT):
            brace = Brace(box, vect, min_num_quads = 5)
            label = TexMobject(label_tex)
            label.next_to(brace, vect, buff = SMALL_BUFF)
            result.add(VGroup(brace, label))
        return result

class WriteDXSquared(Scene):
    def construct(self):
        term = TexMobject("(...)(dx)^2")
        term.set_color(RED)
        self.play(Write(term))
        self.wait()

class MneumonicExample(TeacherStudentsScene):
    def construct(self):
        d, left, right, rp = deriv_q = TexMobject(
            "\\frac{d}{dx}(", "\\sin(x)", "x^2", ")"
        )
        deriv_q.to_edge(UP)

        words = TextMobject(
            "Left ", "d(Right) ", "+", " Right ", "d(Left)",
            arg_separator = ""
        )
        deriv = TexMobject("\\sin(x)", "2x", "+", "x^2", "\\cos(x)")
        for mob in words, deriv:
            VGroup(mob[1], mob[4]).set_color(GREEN)
            mob.next_to(deriv_q, DOWN, buff = MED_LARGE_BUFF)
        deriv.shift(words[2].get_center()-deriv[2].get_center())

        self.add(words)
        self.play(
            Write(deriv_q),
            self.get_teacher().change_mode, "raise_right_hand"
        )
        self.change_student_modes(*["pondering"]*3)

        left_words = VGroup(*words[:2])
        left_terms = VGroup(*deriv[:2])
        self.play(
            left_words.next_to, left_terms, DOWN,
            MED_LARGE_BUFF, RIGHT
        )
        self.play(ReplacementTransform(
            left_words.copy(), left_terms
        ))
        self.wait()
        self.play(*list(map(Indicate, [left, left_words[0], left_terms[0]])))
        self.wait()
        self.play(*list(map(Indicate, [right, left_words[1], left_terms[1]])))
        self.wait()

        right_words = VGroup(*words[2:])
        right_terms = VGroup(*deriv[2:])
        self.play(
            right_words.next_to, right_terms, DOWN,
            MED_LARGE_BUFF, LEFT
        )
        self.play(ReplacementTransform(
            right_words.copy(), right_terms
        ))
        self.wait()
        self.play(*list(map(Indicate, [right, right_words[1], right_terms[1]])))
        self.wait()
        self.play(*list(map(Indicate, [left, right_words[2], right_terms[2]])))
        self.wait(3)

        self.play(self.get_teacher().change_mode, "shruggie")
        self.wait()
        self.change_student_modes(*["confused"]*3)
        self.wait(3)

class ConstantMultiplication(TeacherStudentsScene):
    def construct(self):
        question = TextMobject("What about $\\dfrac{d}{dx}(2\\sin(x))$?")
        answer = TextMobject("2\\cos(x)")
        self.teacher_says(question)
        self.wait()
        self.student_says(
            answer, target_mode = "hooray",
            added_anims = [question.copy().to_edge, UP]
        )
        self.play(self.get_teacher().change_mode, "happy")
        self.change_student_modes("pondering", "hooray", "pondering")
        self.wait(3)

class ConstantMultiplicationFigure(IntroduceProductAsArea):
    CONFIG = {
        "side_func" : lambda x : 1,
        "side_func_label" : "\\text{Constant}",
        "side_func_nudge_label" : "",
        "side_func_derivative" : "",
        "x_unit_to_space_unit" : 3,
        "default_x" : 0.5,
        "dx" : 0.1
    }
    def construct(self):
        self.box_label_group = self.get_box_label_group(self.default_x)
        self.x_slider = self.get_x_slider(self.default_x)
        # df_boxes = self.get_df_boxes()
        # df_box_labels = self.get_df_box_labels(df_boxes)

        self.add(self.box_label_group, self.x_slider)
        self.nudge_x()

class ShoveXSquaredInSine(Scene):
    def construct(self):
        title = TextMobject("Function composition")
        title.to_edge(UP)

        sine = TexMobject("g(", "x", ")", "=", "\\sin(", "x", ")")
        sine.set_color(SINE_COLOR)
        x_squared = TexMobject("h(x)", "=", "x^2")
        x_squared.set_color(X_SQUARED_COLOR)
        group = VGroup(sine, x_squared)
        group.arrange(buff = LARGE_BUFF)
        group.shift(UP)
        composition = TexMobject(
            "g(", "h(x)", ")", "=", "\\sin(", "x^2", ")"
        )
        for i in 0, 2, 4, 6:
            composition[i].set_color(SINE_COLOR)
        for i in 1, 5:
            composition[i].set_color(X_SQUARED_COLOR)
        composition.next_to(group, DOWN, buff = LARGE_BUFF)

        brace = Brace(VGroup(*composition[-3:]), DOWN)
        deriv_q = brace.get_text("Derivative?")

        self.add(group)
        self.play(Write(title))
        self.wait()
        triplets = [
            [sine, (0, 2), (0, 2)],
            [x_squared, (0,), (1,)],
            [sine, (3, 4, 6), (3, 4, 6)],
            [x_squared, (2,), (5,)]
        ]
        for premob, pre_indices, comp_indicies in triplets:
            self.play(*[
                ReplacementTransform(
                    premob[i].copy(), composition[j]
                )
                for i, j in zip(pre_indices, comp_indicies)
            ])
            self.wait()
        self.wait()
        self.play(
            GrowFromCenter(brace),
            Write(deriv_q)
        )
        self.wait()

class ThreeLinesChainRule(ReconfigurableScene):
    CONFIG = {
        "start_x" : 0.5,
        "max_x" : 1,
        "min_x" : 0,
        "top_x" : 3,
        "example_x" : 1.5,
        "dx" : 0.1,
        "line_configs" : [
            {
                "func" : lambda x : x,
                "func_label" : "x",
                "triangle_color" : WHITE,
                "center_y" : 3,
                "x_min" : 0,
                "x_max" : 3,
                "numbers_to_show" : list(range(4)),
                "numbers_with_elongated_ticks" : list(range(4)),
                "tick_frequency" : 0.25,
            },
            {
                "func" : lambda x : x**2,
                "func_label" : "x^2",
                "triangle_color" : X_SQUARED_COLOR,
                "center_y" : 0.5,
                "x_min" : 0,
                "x_max" : 10,
                "numbers_to_show" : list(range(0, 11)),
                "numbers_with_elongated_ticks" : list(range(0, 11, 1)),
                "tick_frequency" : 0.25,
            },
            {
                "func" : lambda x : np.sin(x**2),
                "func_label" : "\\sin(x^2)",
                "triangle_color" : SINE_COLOR,
                "center_y" : -2,
                "x_min" : -2,
                "x_max" : 2,
                "numbers_to_show" : list(range(-2, 3)),
                "numbers_with_elongated_ticks" : list(range(-2, 3)),
                "tick_frequency" : 0.25,
            },
        ],
        "line_width" : 8,
        "triangle_height" : 0.25,
    }
    def construct(self):
        self.introduce_line_group()
        self.draw_function_arrows()
        self.talk_through_movement()
        self.nudge_x()
        self.give_example_of_meaning()

    def introduce_line_group(self):
        self.line_group = self.get_line_group(self.start_x)
        lines, labels = self.line_group

        for line in lines:
            self.play(Write(line, run_time = 2))
        self.wait()
        last_label = labels[0].copy()
        last_label.to_corner(UP+LEFT)
        last_label.set_fill(opacity = 0)
        for label in labels:
            self.play(ReplacementTransform(
                last_label.copy(), label
            ))
            self.wait()
            last_label = label
        for x in self.max_x, self.min_x, self.start_x:
            self.animate_x_change(x, run_time = 1)
        self.wait()

    def draw_function_arrows(self):
        lines, line_labels = self.line_group
        labels = VGroup(*[
            TexMobject("(\\dots)^2").set_color(X_SQUARED_COLOR), 
            TexMobject("\\sin(\\dots)").set_color(SINE_COLOR)
        ])
        arrows = VGroup()
        for lines_subset, label in zip([lines[:2], lines[1:]], labels):
            arrow = Arc(start_angle = np.pi/3, angle = -2*np.pi/3)
            arrow.add_tip()
            arrow.set_color(label.get_color())
            arrow.next_to(VGroup(*lines_subset))
            arrows.add(arrow)
            label.next_to(arrow, RIGHT)

            self.play(
                ShowCreation(arrow),
                Write(label)
            )
            self.wait()
        self.arrows = arrows
        self.arrow_labels = labels

    def talk_through_movement(self):
        lines, labels = self.line_group

        self.animate_x_change(self.top_x, run_time = 4)
        self.wait()
        for label in labels[0], labels[1]:
            oval = Circle(color = YELLOW)
            oval.replace(label, stretch = True)
            oval.scale(2.5)
            oval.move_to(label.get_bottom())
            self.play(ShowCreation(oval))
            self.wait()
            self.play(FadeOut(oval))
        sine_text = TexMobject("\\sin(9) \\approx 0.412")
        sine_text.move_to(labels[-1][-1])
        sine_text.to_edge(DOWN)
        sine_arrow = Arrow(
            sine_text.get_top(),
            labels[-1][0].get_bottom(),
            buff = SMALL_BUFF,
        )
        self.play(
            FadeIn(sine_text),
            ShowCreation(sine_arrow)
        )
        self.wait(2)
        self.play(*list(map(FadeOut, [sine_text, sine_arrow])))
        self.animate_x_change(self.example_x, run_time = 3)

    def nudge_x(self):
        lines, labels = self.line_group
        def get_value_points():
            return [
                label[0].get_bottom()
                for label in labels
            ]
        starts = get_value_points()
        self.animate_x_change(self.example_x + self.dx, run_time = 0)
        ends = get_value_points()
        self.animate_x_change(self.example_x, run_time = 0)

        nudge_lines = VGroup()
        braces = VGroup()
        numbers = VGroup()
        for start, end, line, label, config in zip(starts, ends, lines, labels, self.line_configs):
            color = label[0].get_color()
            nudge_line = Line(start, end)
            nudge_line.set_stroke(color, width = 6)
            brace = Brace(nudge_line, DOWN, buff = SMALL_BUFF)
            brace.set_color(color)
            func_label = config["func_label"]
            if len(func_label) == 1:
                text = "$d%s$"%func_label
            else:
                text = "$d(%s)$"%func_label
            brace.text = brace.get_text(text, buff = SMALL_BUFF)
            brace.text.set_color(color)
            brace.add(brace.text)

            line.add(nudge_line)
            nudge_lines.add(nudge_line)
            braces.add(brace)
            numbers.add(line.numbers)
            line.remove(*line.numbers)
        dx_brace, dx_squared_brace, dsine_brace = braces

        x_value = str(self.example_x)
        x_value_label = TexMobject("=%s"%x_value)
        x_value_label.next_to(labels[0][1], RIGHT)
        dx_squared_value = TexMobject(
            "= 2x\\,dx ", "\\\\ = 2(%s)dx"%x_value
        )
        dx_squared_value.shift(
            dx_squared_brace.text.get_right()+MED_SMALL_BUFF*RIGHT - \
            dx_squared_value[0].get_left()
        )
        dsine_value = TextMobject(
            "$=\\cos(%s)$"%self.line_configs[1]["func_label"],
            dx_squared_brace.text.get_tex_string()
        )
        dsine_value.next_to(dsine_brace.text)
        less_than_zero = TexMobject("<0")
        less_than_zero.next_to(dsine_brace.text)

        all_x_squared_relevant_labels = VGroup(
            dx_squared_brace, dsine_brace,
            labels[1], labels[2],
            dsine_value,
        )
        all_x_squared_relevant_labels.save_state()

        self.play(FadeOut(numbers))
        self.animate_x_change(
            self.example_x + self.dx,
            run_time = 1,
            added_anims = it.chain(
                [GrowFromCenter(dx_brace)],
                list(map(ShowCreation, nudge_lines))
            )
        )
        self.animate_x_change(self.example_x)
        self.wait()
        self.play(Write(x_value_label))
        self.wait()
        self.play(FocusOn(dx_squared_brace))
        self.play(Write(dx_squared_brace))
        self.wiggle_by_dx()
        self.wait()
        for part in dx_squared_value:
            self.play(Write(part))
            self.wait()
        self.play(FadeOut(dx_squared_value))
        self.wait()
        #Needs to be part of everything for the reconfiguraiton
        dsine_brace.set_fill(opacity = 0)
        dsine_value.set_fill(opacity = 0)
        self.add(dsine_brace, dsine_value)
        self.replace_x_squared_with_h()
        self.wait()
        self.play(dsine_brace.set_fill, None, 1)
        self.discuss_dsine_sign(less_than_zero)
        self.wait()
        dsine_value.set_fill(opacity = 1)
        self.play(Write(dsine_value))
        self.wait()
        self.play(
            all_x_squared_relevant_labels.restore,
            lag_ratio = 0.5,
            lag_factor = 3,
            run_time = 3,
        )
        self.__dict__.update(self.__class__.CONFIG)
        self.wait()
        for mob in dsine_value:
            self.play(Indicate(mob))
            self.wait()

        two_x_dx = dx_squared_value[0]
        dx_squared = dsine_value[1]
        two_x_dx_copy = VGroup(*two_x_dx[1:]).copy()
        self.play(FocusOn(two_x_dx))
        self.play(Write(two_x_dx))
        self.play(
            two_x_dx_copy.move_to, dx_squared, LEFT,
            dx_squared.next_to, dx_squared, UP,
            run_time = 2
        )
        self.play(FadeOut(dx_squared))
        for sublist in two_x_dx_copy[:2], two_x_dx_copy[2:]:
            self.play(Indicate(VGroup(*sublist)))
            self.wait()
        self.wait(2)

        self.final_derivative = dsine_value

    def discuss_dsine_sign(self, less_than_zero):
        self.wiggle_by_dx()
        self.wait()
        for x in self.example_x+self.dx, self.example_x:
            self.animate_x_change(x, run_time = 2)
            self.wait()
            if less_than_zero not in self.get_mobjects():
                self.play(Write(less_than_zero))
            else:
                self.play(FadeOut(less_than_zero))

    def replace_x_squared_with_h(self):
        new_config = copy.deepcopy(self.__class__.CONFIG)
        new_config["line_configs"][1]["func_label"] = "h"
        new_config["line_configs"][2]["func_label"] = "\\sin(h)"
        self.transition_to_alt_config(
            return_to_original_configuration = False,
            **new_config
        )

    def give_example_of_meaning(self):
        words = TextMobject("For example,")
        expression = TexMobject("\\cos(1.5^2)\\cdot 2(1.5)\\,dx")
        group = VGroup(words, expression)
        group.arrange(DOWN, aligned_edge = LEFT)
        group.scale(0.8)
        group.to_edge(RIGHT)
        arrow = Arrow(group.get_bottom(), self.final_derivative[0].get_top())

        self.play(*list(map(FadeOut, [self.arrows, self.arrow_labels])))
        self.play(FadeIn(group))
        self.play(ShowCreation(arrow))
        self.wait()
        self.wiggle_by_dx()
        self.wait()


    ########

    def wiggle_by_dx(self, **kwargs):
        kwargs["run_time"] = kwargs.get("run_time", 1)
        kwargs["rate_func"] = kwargs.get("rate_func", there_and_back)
        target_x = self.line_group.x_val + self.dx
        self.animate_x_change(target_x, **kwargs)

    def animate_x_change(self, target_x, **kwargs):
        #Assume fixed lines, only update labels
        kwargs["run_time"] = kwargs.get("run_time", 2)
        added_anims = kwargs.get("added_anims", [])
        start_x = self.line_group.x_val
        def update(line_group, alpha):
            lines, labels = line_group
            new_x = interpolate(start_x, target_x, alpha)
            for line, label, config in zip(lines, labels, self.line_configs):
                new_label = self.get_line_label(
                    line, new_x, **config
                )
                Transform(label, new_label).update(1)
            line_group.x_val = new_x
        self.play(
            UpdateFromAlphaFunc(self.line_group, update),
            *added_anims,
            **kwargs
        )

    def get_line_group(self, x):
        group = VGroup()
        group.lines, group.labels = VGroup(), VGroup()
        for line_config in self.line_configs:
            number_line = self.get_number_line(**line_config)
            label = self.get_line_label(number_line, x, **line_config)
            group.lines.add(number_line)
            group.labels.add(label)
        group.add(group.lines, group.labels)
        group.x_val = x
        return group

    def get_number_line(
        self, center_y, **number_line_config
        ):
        number_line = NumberLine(color = GREY, **number_line_config)
        number_line.stretch_to_fit_width(self.line_width)
        number_line.add_numbers()
        number_line.shift(center_y*UP)
        number_line.to_edge(LEFT, buff = LARGE_BUFF)

        return number_line

    def get_line_label(
        self, number_line, x, func, func_label, triangle_color, 
        **spillover_kwargs
        ):
        triangle = RegularPolygon(
            n=3, start_angle = -np.pi/2,
            fill_color = triangle_color,
            fill_opacity = 0.75,
            stroke_width = 0,
        )
        triangle.set_height(self.triangle_height)
        triangle.move_to(
            number_line.number_to_point(func(x)), DOWN
        )

        label_mob = TexMobject(func_label)
        label_mob.next_to(triangle, UP, buff = SMALL_BUFF, aligned_edge = LEFT)

        return VGroup(triangle, label_mob)

class GeneralizeChainRule(Scene):
    def construct(self):
        example = TexMobject(
            "\\frac{d}{dx}", "\\sin(", "x^2", ")", "=",
            "\\cos(", "x^2", ")", "\\,2x",
        )
        general = TexMobject(
            "\\frac{d}{dx}", "g(", "h(x)", ")", "=",
            "{dg \\over ", " dh}", "(", "h(x)", ")", "{dh \\over", " dx}", "(x)"
        )
        example.to_edge(UP, buff = LARGE_BUFF)
        example.shift(RIGHT)
        general.next_to(example, DOWN, buff = 1.5*LARGE_BUFF)
        for mob in example, general:
            mob.set_color(SINE_COLOR)
            mob[0].set_color(WHITE)
            for tex in "x^2", "2x", "(x)", "{dh", " dx}":
                mob.set_color_by_tex(tex, X_SQUARED_COLOR, substring = True)

        example_outer = VGroup(*example[1:4])
        example_inner = example[2]
        d_example_outer = VGroup(*example[5:8])
        d_example_inner = example[6]
        d_example_d_inner = example[8]

        general_outer = VGroup(*general[1:4])
        general_inner = general[2]
        d_general_outer = VGroup(*general[5:10])
        d_general_inner = general[8]
        d_general_d_inner = VGroup(*general[10:13])

        example_outer_brace = Brace(example_outer)
        example_inner_brace = Brace(example_inner, UP, buff = SMALL_BUFF)
        d_example_outer_brace = Brace(d_example_outer)
        d_example_inner_brace = Brace(d_example_inner, buff = SMALL_BUFF)
        d_example_d_inner_brace = Brace(d_example_d_inner, UP, buff = SMALL_BUFF)

        general_outer_brace = Brace(general_outer)
        general_inner_brace = Brace(general_inner, UP, buff = SMALL_BUFF)
        d_general_outer_brace = Brace(d_general_outer)
        d_general_inner_brace = Brace(d_general_inner, buff = SMALL_BUFF)
        d_general_d_inner_brace = Brace(d_general_d_inner, UP, buff = SMALL_BUFF)

        for brace in example_outer_brace, general_outer_brace:
            brace.text = brace.get_text("Outer")
        for brace in example_inner_brace, general_inner_brace:
            brace.text = brace.get_text("Inner")
        for brace in d_example_outer_brace, d_general_outer_brace:
            brace.text = brace.get_text("d(Outer)")
            brace.text.shift(SMALL_BUFF*LEFT)
        for brace in d_example_d_inner_brace, d_general_d_inner_brace:
            brace.text = brace.get_text("d(Inner)", buff = SMALL_BUFF)

        #d(out)d(in) for example
        self.add(example)
        braces = VGroup(
            example_outer_brace, 
            example_inner_brace, 
            d_example_outer_brace
        )
        for brace in braces:
            self.play(GrowFromCenter(brace))
            self.play(Write(brace.text, run_time = 1))
            self.wait()
        self.wait()
        self.play(*it.chain(*[
            [mob.scale_in_place, 1.2, mob.set_color, YELLOW]
            for mob in (example_inner, d_example_inner)
        ]), rate_func = there_and_back)
        self.play(Transform(
            example_inner.copy(), d_example_inner,
            path_arc = -np.pi/2,
            remover = True
        ))
        self.wait()
        self.play(
            GrowFromCenter(d_example_d_inner_brace),
            Write(d_example_d_inner_brace.text)
        )
        self.play(Transform(
            VGroup(*reversed(example_inner.copy())),
            d_example_d_inner,
            path_arc = -np.pi/2,
            run_time = 2,
            remover = True
        ))
        self.wait()

        #Generalize
        self.play(*list(map(FadeIn, general[:5])))
        self.wait()
        self.play(
            Transform(example_outer_brace, general_outer_brace),
            Transform(example_outer_brace.text, general_outer_brace.text),
            Transform(example_inner_brace, general_inner_brace),
            Transform(example_inner_brace.text, general_inner_brace.text),
        )
        self.wait()
        self.play(
            Transform(d_example_outer_brace, d_general_outer_brace),
            Transform(d_example_outer_brace.text, d_general_outer_brace.text),
        )
        self.play(Write(d_general_outer))
        self.wait(2)
        self.play(
            Transform(d_example_d_inner_brace, d_general_d_inner_brace),
            Transform(d_example_d_inner_brace.text, d_general_d_inner_brace.text),
        )
        self.play(Write(d_general_d_inner))
        self.wait(2)

        #Name chain rule
        name = TextMobject("``Chain rule''")
        name.scale(1.2)
        name.set_color(YELLOW)
        name.to_corner(UP+LEFT)
        self.play(Write(name))
        self.wait()

        #Point out dh bottom
        morty = Mortimer().flip()
        morty.to_corner(DOWN+LEFT)
        d_general_outer_copy = d_general_outer.copy()
        morty.set_fill(opacity = 0)
        self.play(
            morty.set_fill, None, 1,
            morty.change_mode, "raise_left_hand",
            morty.look, UP+LEFT,
            d_general_outer_copy.next_to, 
            morty.get_corner(UP+LEFT), UP, MED_LARGE_BUFF,
            d_general_outer_copy.shift_onto_screen
        )
        self.wait()        
        circle = Circle(color = YELLOW)
        circle.replace(d_general_outer_copy[1])
        circle.scale_in_place(1.4)
        self.play(ShowCreation(circle))
        self.play(Blink(morty))
        self.wait()
        inner = d_general_outer_copy[3]
        self.play(
            morty.change_mode, "hooray",
            morty.look_at, inner,
            inner.shift, UP
        )
        self.play(inner.shift, DOWN)
        self.wait()
        self.play(morty.change_mode, "pondering")
        self.play(Blink(morty))
        self.wait()
        self.play(*list(map(FadeOut, [
            d_general_outer_copy, inner, circle
        ])))

        #Show cancelation
        braces = [
            d_example_d_inner_brace,
            d_example_outer_brace,
            example_inner_brace,
            example_outer_brace,
        ]
        texts = [brace.text for brace in braces]
        self.play(*list(map(FadeOut, braces+texts)))

        to_collapse = VGroup(VGroup(*general[7:10]), general[12])
        dg_dh = VGroup(*general[5:7])
        dh_dx = VGroup(*general[10:12])
        to_collapse.generate_target()
        points = VGroup(*list(map(VectorizedPoint, 
            [m.get_left() for m in to_collapse]
        )))
        self.play(
            Transform(to_collapse, points),
            dh_dx.next_to, dg_dh,
            morty.look_at, dg_dh,
        )
        self.wait()
        for mob in list(dg_dh)+list(dh_dx):
            circle = Circle(color = YELLOW)
            circle.replace(mob)
            circle.scale_in_place(1.3)
            self.play(ShowCreation(circle))
            self.wait()
            self.play(FadeOut(circle))

        strikes = VGroup()
        for dh in dg_dh[1], dh_dx[0]:
            strike = TexMobject("/")
            strike.stretch(2, dim = 0)
            strike.rotate(-np.pi/12)
            strike.move_to(dh)
            strike.set_color(RED)
            strikes.add(strike)
        self.play(Write(strikes))
        self.play(morty.change_mode, "hooray")
        equals_dg_dx = TexMobject("= \\frac{dg}{dx}")
        equals_dg_dx.next_to(dh_dx)
        self.play(Write(equals_dg_dx))
        self.play(Blink(morty))
        self.wait(2)

        ##More than a notational trick
        self.play(
            PiCreatureSays(morty, """
                This is more than a
                notational trick
            """),
            VGroup(
                dg_dh, dh_dx, equals_dg_dx, strikes,
                *general[:5]
            ).shift, DOWN,
            FadeOut(example)
        )
        self.wait()
        self.play(Blink(morty))
        self.wait()

class WatchingVideo(PiCreatureScene):
    def construct(self):
        laptop = Laptop()
        laptop.scale(2)
        laptop.to_corner(UP+RIGHT)
        randy = self.get_primary_pi_creature()
        randy.move_to(laptop, DOWN+LEFT)
        randy.shift(MED_SMALL_BUFF*UP)
        randy.look_at(laptop.screen)


        formulas = VGroup(*[
            TexMobject("\\frac{d}{dx}\\left( %s \\right)"%s)
            for s in [
                "e^x \\sin(x)",
                "\\sin(x) \\cdot  \\frac{1}{\\cos(x)}",
                "\\cos(3x)^2",
                "e^x(x^2 + 3x + 2)",
            ]
        ])
        formulas.arrange(
            DOWN, 
            buff = MED_LARGE_BUFF,
            aligned_edge = LEFT
        )
        formulas.next_to(randy, LEFT, buff = MED_LARGE_BUFF)
        formulas.shift_onto_screen()

        self.add(randy, laptop)
        self.wait()
        self.play(randy.change_mode, "erm")
        self.play(Blink(randy))
        self.wait()
        self.play(randy.change_mode, "maybe")
        self.wait()
        self.play(Blink(randy))
        for formula in formulas:
            self.play(
                Write(formula, run_time = 2),
                randy.change_mode, "thinking"
            )
            self.wait()

    def create_pi_creatures(self):
        return [Randolph().shift(DOWN+RIGHT)]

class NextVideo(TeacherStudentsScene):
    def construct(self):
        series = VideoSeries()
        series.to_edge(UP)
        next_video = series[4]

        pre_expression = TexMobject(
            "x", "^2", "+", "y", "^2", "=", "1"
        )
        d_expression = TexMobject(
            "2", "x", "\\,dx", "+", "2", "y", "\\,dy", "=", "0"
        )
        expression_to_d_expression_indices = [
            1, 0, 0, 2, 4, 3, 3, 5, 6
        ]
        expression = VGroup()
        for i, j in enumerate(expression_to_d_expression_indices):
            submob = pre_expression[j].copy()
            if d_expression.expression_parts[i] == "2":
                two = TexMobject("2")
                two.replace(submob)
                expression.add(two)
            else:
                expression.add(submob)

        for mob in expression, d_expression:
            mob.scale(1.2)
            mob.next_to(
                self.get_teacher().get_corner(UP+LEFT), UP,
                buff = MED_LARGE_BUFF
            )
            mob.shift_onto_screen()

        axes = Axes(x_min = -3, x_max = 3, color = GREY)
        axes.add(Circle(color = YELLOW))
        line = Line(np.sqrt(2)*UP, np.sqrt(2)*RIGHT)
        line.scale_in_place(1.5)
        axes.add(line)

        axes.scale(0.5)
        axes.next_to(d_expression, LEFT)

        self.add(series)        
        self.play(
            next_video.shift, 0.5*DOWN,
            next_video.set_color, YELLOW,
            self.get_teacher().change_mode, "raise_right_hand"
        )
        self.wait()
        self.play(
            Write(expression),
            *[
                ApplyMethod(pi.change_mode, "pondering")
                for pi in self.get_students()
            ]
        )
        self.play(FadeIn(axes))
        self.wait()
        self.remove(expression)
        self.play(Transform(expression, d_expression, path_arc = np.pi/2))
        self.wait()
        self.play(
            Rotate(
                line, np.pi/4, 
                about_point = axes.get_center(),
                rate_func = wiggle,
                run_time = 3
            )
        )
        self.wait(2)

class Chapter4Thanks(PatreonThanks):
    CONFIG = {
        "specific_patrons" : [
            "Ali  Yahya",
            "Meshal  Alshammari",
            "CrypticSwarm    ",
            "Ankit   Agarwal",
            "Yu  Jun",
            "Shelby  Doolittle",
            "Dave    Nicponski",
            "Damion  Kistler",
            "Juan    Benet",
            "Othman  Alikhan",
            "Justin Helps",
            "Markus  Persson",
            "Dan Buchoff",
            "Derek   Dai",
            "Joseph  John Cox",
            "Luc Ritchie",
            "Nils Schneider",
            "Mathew Bramson",
            "Guido   Gambardella",
            "Jerry   Ling",
            "Mark    Govea",
            "Vecht",
            "Shimin Kuang",
            "Rish    Kundalia",
            "Achille Brighton",
            "Kirk    Werklund",
            "Ripta   Pasay",
            "Felipe  Diniz",
        ],
        "patron_group_size" : 8,
    }

class Thumbnail(IntroduceProductAsArea):
    CONFIG = {
        "default_x" : 0.8,
        "dx" : 0.05
    }
    def construct(self):
        self.x_slider = self.get_x_slider(self.default_x)
        blg = self.box_label_group = self.get_box_label_group(
            self.default_x
        )
        df_boxes = self.get_df_boxes()
        df_boxes.space_out_submobjects(1.1)
        df_boxes.move_to(blg[0], UP+LEFT)
        blg[1][1].next_to(df_boxes[-1], RIGHT)
        df_box_labels = self.get_df_box_labels(df_boxes)
        blg.add(df_boxes, df_box_labels)
        blg.set_height(FRAME_HEIGHT-2*MED_LARGE_BUFF)
        blg.center()
        self.add(blg)