AP Calculus BC Unit 3: Differentiation: Composite, Implicit, and Inverse Functions Notes

Introduction

Nested expressions multiply local rates through the chain rule, while products and quotients retain their outer structural rules.

Implicit relations yield tangent slopes without a global y-formula, and inverse derivatives use reciprocal slopes at corresponding reflected points.

Higher derivatives form new functions with their own domains, units, zeros, and interpretations.

Current exam guidance: May 2027 format: 42 multiple-choice questions and 6 free-response questions. Calculator availability changes by part; series, parametric, polar, and vector reasoning remain part of the BC review.

3.1

The Chain Rule

Why do derivatives multiply when one function is nested inside another?

Short answer

A change in x first changes the inner output, then the outer function scales that inner change; the local scale factors multiply.

Know firstComposition · Power rule · Derivative notation

Differentiate layers from outside to inside

For y=F(g(x)), a small x-change produces approximately g′(x)Δx units of inner change. The outer function responds at rate F′ evaluated at g(x). Multiplying those two local rates gives the total response.

In (1+x²)^4, the outer fourth power contributes 4(1+x²)^3 and the inner expression contributes 2x. Their product is 8x(1+x²)^3. Keep the unchanged inner expression inside the outer derivative.

Formula

Formula and conditions

\frac{d}{dx}F(g(x))=F\prime(g(x))g\prime(x)

Outer rate at the current inner value times inner rate.

Use when: Both component derivatives exist at the corresponding inputs.
Do not assume: Differentiating only the outer layer loses g′(x).

Nested change produces an outer and inner factor

The even function (1+x²)^4 has the odd derivative 8x(1+x²)^3. The factor 2x records how quickly the inner x² changes.

The interactive version of this figure is part of the full study guide.

Nested change produces an outer and inner factor: values

Case	Result
Outer derivative	$4(1+x^{2})^{3}$
Inner derivative	2x
Product	$8x(1+x^{2})^{3}$

Keep the derivative structure visible

Mark the outermost function.
Differentiate it while keeping its input unchanged.
Multiply by the derivative of that input.
Repeat for additional nested layers.

Keep these three ideas

Identify layers.
Outer derivative keeps inner input.
Multiply every inner rate.

Keep reading Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

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3.1 The Chain RuleRead

3.2 Implicit Differentiation

3.3 Differentiating Inverse Functions

3.4 Differentiating Inverse Trigonometric Functions

3.5 Selecting Procedures for Calculating Derivatives

3.6 Calculating Higher-Order Derivatives

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