3.1 The Chain Rule

Learning objectives

—FUN-3.C—Calculate derivatives of compositions of differentiable functions• FUN-3.C Calculate derivatives of compositions of differentiable functions.• FUN-3.C.1 The chain rule provides a way to differentiate composite functions.• Enduring understanding FUN-3: Recognizing opportunities to apply derivative rules can simplify differentiation.

A composite function has the form $y=f(g(x))$ : the inner function $g$ produces the input for the outer function $f$ . The total derivative multiplies the rate through both stages.

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)\qquad\text{or}\qquad\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}

Identify the inner expression $u=g(x)$ .
Differentiate the outer function with respect to $u$ , leaving $u$ in place.
Multiply by $du/dx$ .
Substitute the original inner expression and simplify.

For $y=(3x^2-1)^5$ , let $u=3x^2-1$ . Then $dy/du=5u^4$ and $du/dx=6x$ , so $\frac{dy}{dx}=5(3x^2-1)^4(6x)=30x(3x^2-1)^4.$

Do not stop after differentiating only the outer function: $5(3x^2-1)^4$ is missing the inner derivative $6x$ . The extra factor is unnecessary only when the inner function is exactly $x$ , whose derivative is $1$ .

Next Topic

—3.2 Implicit DifferentiationRead