The chain rule

A function nested inside another is differentiated with the chain rule¹. We check it with $f(x) = \sin(x^2)$.

The formula

Differentiate the outside with respect to the inside, and multiply by the derivative of the inside with respect to $x$.

Here $g(t) = \sin t$ and $h(x) = x^2$, so $g'(t) = \cos t$ and $h'(x) = 2x$, and the derivative is as follows.

The factor $\cos(x^2)$ alone is not enough. The derivative $2x$ of the inside always multiplies it.

How the $2x$ shows itself

The graph makes the effect of $2x$ visible. The larger $x$ becomes, the faster the inside $x^2$ grows, so the wave of $\sin(x^2)$ bunches up toward the right. The amplitude of the derivative also grows in proportion to $2x$.

The spacing of the wave can be followed numerically as well. We have $f = 0$ when $x^2 = n\pi$, that is at $x = \sqrt{n\pi}$.

It nests to any depth

For $\sin\left( e^{x^2} \right)$, multiplying from the outside inward by $\cos\left( e^{x^2} \right)$, then $e^{x^2}$, then $2x$ gives $2xe^{x^2}\cos\left( e^{x^2} \right)$.

A way to remember it

It looks as though $dt$ cancels like a fraction, but that is only the appearance of the notation; the proof goes by rewriting it as a product of average rates of change.

Relation to substitution in integrals

Integration by substitution is this rule used backwards. Knowing that the chain rule multiplies by the derivative of the inside, one can read off the following integral.

What differentiation multiplies in, integration has to find and take back out.

The wave bunching up to the right is $y = \sin(x^2)$, the wave of growing amplitude is the derivative, and the large dots are the origin and $\left( \sqrt{\pi}, 0 \right)$.

1. Chain rule, Wikipedia