3.1 The Chain Rule

The rule

For a composite function f(g(x)): d/dx[f(g(x))] = f'(g(x))·g'(x). In words: the derivative of the outside (leaving the inside alone) times the derivative of the inside.

Examples

d/dx[(x²+1)³] = 3(x²+1)²·(2x) = 6x(x²+1)². d/dx[sin(2x)] = cos(2x)·2 = 2cos(2x). d/dx[e^(3x)] = 3e^(3x).

Key terms

Chain rule

d/dx[f(g(x))] = f'(g(x))·g'(x).

In words

Derivative of outside times derivative of inside.

When to use

Composite (nested) functions.

Example

d/dx[(x²+1)³] = 6x(x²+1)².

Trig example

d/dx[sin(2x)] = 2cos(2x).

Try an AP-style question

d/dx sin(x²) = ?

1. cos(x²)
2. 2x cos(x²)
3. 2x sin(x²)
4. cos(2x)