The Chain Rule

By the end of this lesson, you will be able to apply the chain rule to differentiate composite functions and identify when it is needed.

Composite Functions

A composite function $f(g(x))$ is a function-inside-a-function. The chain rule is the tool for differentiating them.

Examples of composites: $\sin(x^2)$, $(x^3+1)^5$, $e^{\tan x}$, $\ln(\cos x)$

For each, identify:

• Outer function $f$: the "outside" operation
• Inner function $g$: the "inside" argument

The Chain Rule

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Memory: "Derivative of the outside (evaluated at the inside), times derivative of the inside."

Leibniz notation: If $y = f(u)$ and $u = g(x)$:

$$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$

Examples

Example 1: $\frac{d}{dx}[\sin(x^2)]$

• Outer: $\sin(u)$; $f'(u) = \cos(u)$
• Inner: $u = x^2$; $g'(x) = 2x$

$= \cos(x^2) \cdot 2x = 2x\cos(x^2)$

Example 2: $\frac{d}{dx}[(3x^2 + 1)^5]$

• Outer: $u^5$; $f'(u) = 5u^4$
• Inner: $u = 3x^2 + 1$; $g'(x) = 6x$

$= 5(3x^2+1)^4 \cdot 6x = 30x(3x^2+1)^4$

Example 3: $\frac{d}{dx}[e^{\tan x}]$

• Outer: $e^u$; $f'(u) = e^u$
• Inner: $u = \tan x$; $g'(x) = \sec^2 x$

$= e^{\tan x} \cdot \sec^2 x$

Example 4: $\frac{d}{dx}[\ln(\cos x)]$

• Outer: $\ln(u)$; $f'(u) = 1/u$
• Inner: $u = \cos x$; $g'(x) = -\sin x$

$= \frac{1}{\cos x} \cdot (-\sin x) = -\tan x$

Nested Chains (Triple Composition)

$\frac{d}{dx}[\sin^3(2x)] = ?$

This is $[\sin(2x)]^3$:

• Outer: $u^3$; derivative: $3u^2$
• Middle: $\sin(v)$; derivative: $\cos(v)$
• Inner: $v = 2x$; derivative: 2

Chain rule applied twice:

$= 3[\sin(2x)]^2 \cdot \cos(2x) \cdot 2 = 6\sin^2(2x)\cos(2x)$