Differentiate composite functions using the chain rule

The chain rule expresses how the rate of change of a composite function depends on both the outer function’s derivative and the inner function’s derivative: $\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$ . This includes identifying inner and outer functions and differentiating common algebraic, exponential, logarithmic, and trigonometric composites without omitting the inner derivative; more advanced generalizations, such as multivariable chain rules, are not included.

Detailed Explanation: 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.

Learn by doing: Differentiate composite functions using the chain rule

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Derivative Rules - Multi Chain Natural and General Exponential to Derivative

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