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The chain rule

The chain rule differentiates a function wrapped around another function. You differentiate the outer function, leave the inner function in place, and multiply by the derivative of the inner function. For a power of a linear inner function both the old exponent and the slope of the inner function appear as factors. The exponential function with base e is its own derivative, so a coefficient in the exponent becomes a factor in front.

((ax+b)^n)' = na(ax+b)^{n-1}

power of a linear inner function

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

exponential

Symbols

$a$	slope of the inner function
$k$	coefficient in the exponent

Example

$f(x) = (2x - 1)^5$ gives $f'(x) = 10(2x - 1)^4$ , so $f'(1) = 10$ .

$(e^{3x})' = 3e^{3x}$ , which is $3$ when $x = 0$ .

Outer derivative times inner derivative.

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Part of Mathematics S2: Differentiation rules in practice.