The Chain Rule

None of §7's five formulae, on their own, can differentiate a composite function — a function of a function, such as $e^{3x^2+1}$ or $\ln(5x-2)$, where the "outer" function ($e^{(\cdot)}$ or $\ln(\cdot)$) is applied not to $x$ directly but to another function of $x$. The chain rule is the tool for exactly this situation.

Method:

1. Identify the "outer" function and the "inner" function — write $y=f(u)$ where $u$ is whatever expression the outer function is applied to.
2. Differentiate the outer function with respect to $u$, using the standard formulae of §7, to get $\dfrac{dy}{du}$.
3. Differentiate the inner function $u$ with respect to $x$ to get $\dfrac{du}{dx}$.
4. Multiply the two results together, then substitute back so the final answer is expressed in $x$ alone (not $u$).

For example, to differentiate $y = e^{3x^2+1}$: the outer function is $e^{(\cdot)}$ applied to $u = 3x^2+1$; $\dfrac{dy}{du}=e^u$ and $\dfrac{du}{dx}=6x$, so $\dfrac{dy}{dx} = e^u \cdot 6x = 6x\,e^{3x^2+1}$.