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Chain rule

This original textbook reading covers part of the AP topic. Follow the AP course outline for complete topic instruction and practice.

For any function $F (x) = f (g (h (\dots (x))))$ ,

$F^{'} (x) = f^{'} (g (h (\dots (x))) \cdot g^{'} (h (\dots (x))) \cdot (h^{'} (\dots (x))$

Example 1: $f (x) = \sin (e^{x})$

$f^{'} (x) = \cos (e^{x}) \cdot e^{x}$

Example 2: $f (x) = e^{\sin x}$

$f^{'} (x) = e^{\sin x} \cdot \cos x$

Related Glossary Terms

Implicit differentiation

Index

Chapter 4 - The Chain Rule

Chapter 4 - The Chain Rule

Chapter 4 - The Chain Rule

Chapter 4 - FFR - Implicit Differentiation

Chapter 4 - FFR - Implicit Differentiation

Chapter 4 - FFR - Implicit Differentiation

Chapter 5 - Powers and Logarithms

Chapter 5 - Inverse Functions

Chapter 8 - Inverses Revisited

Glossary and deeper reading · 1 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

Chain rule
The chain rule differentiates a composition by multiplying the outer derivative at the inner output by the inner derivative. For differentiable functions, (f(g(x)))′=f′(g(x))g′(x). For example, the derivative of sin(x²) is cos(x²)·2x. Nested compositions require a factor for each successive layer.
Read more: 3.1 · Chain rule →

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