Summary AP Calculus AB Unit 3: Differentiation: Composite, Implicit, and Inverse Functions - Study Guide

AP Calculus AB Unit 3: Composite, Implicit, & Inverse



Differentiation II
Composite, Implicit, and Inverse Functions




1. The Chain Rule

The Chain Rule is used for differentiating composite functions (functions inside other functions), typ-
ically written as f (g(x)).

Chain Rule Definition
If y = f (u) and u = g(x), then:
dy dy du
= ·
dx du dx
d
[f (g(x))] = f ′ (g(x)) · g ′ (x)
dx
"Derivative of the outer function (evaluated at the inner) × derivative of the inner function."

Example: Let y = (3x2 + 1)5 .

• Outer: (. . . )5 → 5(. . . )4

• Inner: 3x2 + 1 → 6x

• Result: 5(3x2 + 1)4 · (6x) = 30x(3x2 + 1)4


2. Implicit Differentiation

Used when y cannot be easily isolated (e.g., x2 + y 2 = 25).

Key Formula

Steps for Implicit Differentiation:

1. Differentiate both sides with respect to x.
dy
2. Whenever you differentiate a y term, multiply by dx (Chain Rule).
dy
3. Collect all terms with dx on one side.
dy
4. Factor out dx and solve for it.


Strategy Tip – Watch out for Product Rule mixed with Implicit Differentiation! Example:
d dy
dx (xy) = (1)(y) + (x)( dx ).




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