Chain Rule

Many functions that arise in practice are not simple polynomials or single trigonometric

expressions, but composite functions -- a function applied to the output of another

function, such as $y=\sin(3x^2+2x)$ or $y=(2x^3-5x+1)^6$. Differentiating a composite function

needs a rule of its own: the chain rule.

Statement. Let $y=f(u)$ where $u=g(x)$, and suppose $g$ is differentiable at $x$ and $f$ is

differentiable at $u=g(x)$. Then the composite function $y=f(g(x))$ is differentiable at $x$, and

Why it works. Write the increment form: if $x$ changes by a small amount $\Delta x$, this

produces a corresponding change $\Delta u$ in $u=g(x)$, which in turn produces a change $\Delta y$

in $y=f(u)$. Provided $\Delta u\neq0$,

As $\Delta x\to0$, differentiability of $g$ at $x$ (Section 2) makes $g$ continuous there, so

$\Delta u\to0$ as well; then $\Delta y/\Delta u\to dy/du$ (since $f$ is differentiable at $u$) and

$\Delta u/\Delta x\to du/dx$ (since $g$ is differentiable at $x$), giving $dy/dx=(dy/du)(du/dx)$

in the limit. (A fully rigorous proof must separately handle the case $\Delta u=0$ for some

values of $\Delta x$ arbitrarily close to $0$; this technical point is set aside at this level,

and the chain rule is used exactly as stated above.)

Practical recipe. To differentiate $y=f(g(x))$: (i) differentiate the outer function $f$

with respect to its own argument, leaving the inner function $g(x)$ untouched inside; (ii)

multiply by the derivative of the inner function $g(x)$. Symbolically,

Chains of more than two functions. The rule extends to any number of nested functions: if

$y=f(u)$, $u=g(v)$, $v=h(x)$, then

This is used, for instance, to differentiate $y=\sin\big(\sqrt{x^2+1}\big)$, where the outer

function is $\sin$, the middle function is a square root, and the inner function is $x^2+1$.

The chain rule underlies almost every later section of this chapter: the derivatives of the

inverse trigonometric functions (Section 4), of $e^{f(x)}$ and $\ln f(x)$ (Section 7), and the …