Chain Rule Visualizer – AP Calculus AB

Learning Objectives

Inner Function $g(x)$: Drives the intermediary variable $u$, like the turning of a steering wheel turning an axle.

Outer Function $f(u)$: Takes $u$ and converts it to the final output $y$, like the axle turning the car's wheels.

The Multiplier: The total rate of change $\frac{dy}{dx}$ is the product of how fast the outer responds to the inner ($\frac{dy}{du}$) multiplied by how fast the inner responds to x ($\frac{du}{dx}$).

Tags

DerivativesChain RuleComposite Functions

       COMPOSITE FUNCTION

       $y = f(g(x)) = \sin(x^2)$

Input Driver (x)

Velocity $\frac{dx}{dt}$ = 1.0 rev/s

The Chain Mechanism

$\frac{dy}{dx} = f'(g(x)) \cdot g'(x)$

COMPONENT BREAKDOWN

Inner: $u = g(x) = x^2$

$\frac{du}{dx} = 2x$

Outer: $y = f(u) = \sin(u)$

$\frac{dy}{du} = \cos(u) = \cos(x^2)$

Total $\frac{dy}{dx} = \cos(x^2) \cdot 2x$

Mechanical Analogy: Think of composite functions like interconnected gears. Gear A (x) turns Gear B (u), which in turn turns Gear C (y). To find out how fast Gear C spins relative to Gear A, you MULTIPLY their connection ratios!

AP Exam Tip

Always peel the onion from the OUTSIDE in. Derive the outer function, LEAVE the inside completely alone, then multiply by the derivative of the inside. Never forget the $\cdot \ g'(x)$ tail!