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The chain rule

The chain rule is used when one function sits inside another. You differentiate the outer function, leave the inner function untouched, and multiply by the derivative of the inner function. For a power, the outer derivative is the exponent times the power with degree one lower. The rule also covers $e^{kx}$ and $\ln(u)$ , where the inner derivative always appears as an extra factor.

(u^n)' = n u^{n-1}\cdot u'

chain rule for a power

(e^{kx})' = ke^{kx}

chain rule for an exponential

Symbols

$u$	the inner function
$n$	the exponent
$k$	the coefficient

Example

$f(x) = (2x + 1)^3$ gives $f'(x) = 3(2x + 1)^2 \cdot 2 = 6(2x + 1)^2$ . Then $f'(0) = 6$ and $f'(1) = 54$ .

If you forget to multiply by the inner derivative, the factor 2 is missing in this example.

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Part of Mathematics R1: Differentiation and curve analysis.