Chain Rule

Level 1 - Math I (Physics) topic page in Differentiation.

The chain rule is used to differentiate composite functions.

If \(y = f(g(x))\), then:

Chain Rule

\[\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\]

where \(u = g(x)\).

Alternative Form

Chain Rule Alt

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

Find \(\frac{d}{dx}[\sin(x^2)]\)

Let \(u = x^2\), so \(f(u) = \sin(u)\)

Example

\[\frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot 2x = 2x\cos(x^2)\]

For \(y = [g(x)]^n\):

Power Chain

\[\frac{dy}{dx} = n[g(x)]^{n-1} \cdot g'(x)\]