The chain rule: derivatives of composite functions - Derivatives from scratch

The chain rule

A composite function is a function inside a function, for example $(3x + 1)^5$ : on the outside there’s a power, on the inside there’s $3x + 1$ . You differentiate it like this:

\big(f(g(x))\big)' = f'(g(x)) \cdot g'(x)

In words: the derivative of the outer function (with the inside left untouched) times the derivative of the inner function.

Example 1: a power

For $(3x + 1)^5$ , the outer function is $u^5$ and the inner function is $u = 3x + 1$ :

\big((3x + 1)^5\big)' = 5(3x + 1)^4 \cdot 3 = 15(3x + 1)^4

Example 2: sine

\big(\sin(x^2)\big)' = \cos(x^2) \cdot 2x

Example 3: a square root

\left(\sqrt{1 + x^2}\right)' = \frac{1}{2\sqrt{1 + x^2}} \cdot 2x = \frac{x}{\sqrt{1 + x^2}}

A common mistake

Forgetting the inner derivative. If you skipped multiplying by $3$ in Example 1, you’d get $5(3x + 1)^4$ - a result three times too small. A good habit: once you’ve differentiated the outer function, always ask yourself what’s inside.