The Chain Rule (Composite Functions)

A composite function is a "function of a function", such as $(3x^2+5)^4$ or $\sin(x^2)$ or $\log_e(1+x^2)$ — an outer function applied to an inner expression. The chain rule differentiates it.

If $y$ is a function of $u$, and $u$ is a function of $x$, then

In words: differentiate the outer function (keeping the inside unchanged), then multiply by the derivative of the inside.

Worked illustration — $y = (3x^2+5)^4$. Put $u = 3x^2+5$, so $y = u^4$. Then $\dfrac{dy}{du} = 4u^3$ and $\dfrac{du}{dx} = 6x$. Therefore

With practice the substitution $u$ is done mentally: "bring down the power $4$, reduce it to $3$, keep the bracket, then multiply by the bracket's derivative $6x$."

Combining with standard functions: $\dfrac{d}{dx}\,e^{g(x)} = e^{g(x)}\,g'(x)$, $\ \dfrac{d}{dx}\,\sin(g(x)) = \cos(g(x))\,g'(x)$, $\ \dfrac{d}{dx}\,\log_e(g(x)) = \dfrac{g'(x)}{g(x)}$ — in each case, the standard derivative of the outer function, times the derivative of the inside.