Skill: Differentiate composite functions using the chain rule

When one function is inside another, use the chain rule:

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

Think of the expression as having:

• an inner function: the function inside parentheses
• an outer function: the function acting on the inner function

Example

Differentiate

$$y=(3x^2-5)^4$$

Step 1: Identify the inner and outer functions.

The inner function is

$$u=3x^2-5$$

The outer function is

$$y=u^4$$

Step 2: Differentiate the outer function.

Using the power rule,

$$\frac{dy}{du}=4u^3$$

Step 3: Differentiate the inner function.

$$\frac{du}{dx}=6x$$

Step 4: Multiply the derivatives.

The chain rule gives

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

Substitute the derivatives:

$$\frac{dy}{dx}=4u^3\cdot 6x$$

Replace $u$ with $3x^2-5$:

$$\frac{dy}{dx}=4(3x^2-5)^3\cdot 6x$$

Therefore,

$$\boxed{\frac{dy}{dx}=24x(3x^2-5)^3}$$

The factor $6x$ is the derivative of the inner function. It must be included; leaving it out would mean the chain rule was not fully applied.