Chain Rule

Reading

• Sections 3.7

Practice problems

• Section 3.7: 7, 9, 13, 21, 27, 31, 69
• To turn in (together with 3.6): 3.7 12, 70

Notes

The Chain Rule

The chain rule tells us how to differentiate the composite of two functions:

Composite function \[(f \circ g)(x) = f(g(x))\]

We evaluate the composite function on an \(x\) by applying the inner function first, followed by the outer function.

\[(f(g(x)))' = f'(g(x))g'(x)\] “Derivative of the outer, applied to the inner, times the derivative of the inner”

Examples:

• \(\sin(x^3)\). Here \(f(u) = \sin u\), \(u=g(x) = x^3\). Derivative is \(\cos(x^3)3x^2\).
• \(\tan\left(\frac{1}{1+x}\right)\).
• \(\cos(\cos x)\).

Alternative formulation:

If \(u=g(x)\), and \(y=f(u)\), we can view \(y=f(g(x))\) as a function of \(x\). Then: \[\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}\]

Special case of chain rule:

General Power Rule \[\frac{d}{dx}g(x)^n = n g(x)^{n-1}g'(x)\]