The Chain Rule, but Actually Intuitive

The rule (and why it looks confusing)

Most textbooks state the chain rule like this:

$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$

That notation is elegant once you understand it, but completely opaque when you first see it. Let's build the intuition first.

Intuition: peeling an onion

Think of a composite function as an onion. For example:

$y = (3x^2 + 1)^5$

The outer layer is $(\ldots)^5$ — the "raise to the power of 5" operation.
The inner layer is $3x^2 + 1$ — what's actually inside.

The chain rule says:

Differentiate the outside (leaving the inside untouched),
then multiply by the derivative of the inside.

$\frac{dy}{dx} = 5(3x^2 + 1)^4 \cdot 6x = 30x(3x^2 + 1)^4$

Worked examples

Example 1 — Power of a function

Differentiate $y = (2x^3 - 4)^7$.

Let $u = 2x^3 - 4$, so $y = u^7$.

$\frac{dy}{du} = 7u^6, \qquad \frac{du}{dx} = 6x^2$

$\therefore \quad \frac{dy}{dx} = 7(2x^3-4)^6 \cdot 6x^2 = 42x^2(2x^3-4)^6$

Example 2 — Trig

Differentiate $y = \sin(x^2 + 1)$.

Outer: $\sin(\ldots)$ → derivative $\cos(\ldots)$
Inner: $x^2 + 1$ → derivative $2x$

$\frac{dy}{dx} = \cos(x^2 + 1) \cdot 2x = 2x\cos(x^2 + 1)$

Example 3 — Exponential

Differentiate $y = e^{3x}$.

Outer: $e^{(\ldots)}$ → derivative $e^{(\ldots)}$
Inner: $3x$ → derivative $3$

$\frac{dy}{dx} = e^{3x} \cdot 3 = 3e^{3x}$

A note on the Leibniz notation

The fraction-like notation $\frac{dy}{du} \cdot \frac{du}{dx}$ is helpful because it looks like the $du$'s cancel:

$\frac{dy}{\cancel{du}} \cdot \frac{\cancel{du}}{dx} = \frac{dy}{dx}$

This isn't literally true (derivatives are limits, not fractions), but as a memory device it's almost impossible to forget.

Common mistakes

The mistake in every case: forgetting to multiply by the inner derivative.

Practice

1. Differentiate $y = (x^2 - 3x)^4$
2. Differentiate $y = \cos(5x^2)$
3. Differentiate $y = e^{x^3 - 2}$
4. Differentiate $y = \ln(4x^2 + 1)$ (recall: $\frac{d}{dx}[\ln u] = \frac{1}{u}\cdot\frac{du}{dx}$)