the Chain Rule?

The rule for differentiating a function inside another function — multiply the derivative of the outside by the derivative of the inside.

Simple Definition

Many functions in calculus are compositions — one function nested inside another. For example, $\sin(x^2)$ is the sine function applied to $x^2$. To differentiate such a function, you can't just differentiate the outer part and ignore the inner — you need the chain rule.

The chain rule says: differentiate the outer function, keep the inner function unchanged, then multiply by the derivative of the inner function. That final multiplication is what most students forget — and why the chain rule is the most missed rule on calculus exams.

Intuition

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Gears turning gears

Imagine two gears connected. When the small gear turns, it drives the large gear. The rate at which the large gear turns depends on both the rate of the small gear and the gear ratio between them. The chain rule works the same way: the outer rate of change is scaled by the inner rate of change.

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A real-world chain

Your energy level depends on your blood sugar. Your blood sugar depends on what you ate. If you want to know how energy changes with food, you multiply: (how energy changes with blood sugar) × (how blood sugar changes with food). That's the chain rule — a chain of rates.

Formal Definition

Chain Rule — Leibniz Notation

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

Chain Rule — Function Notation

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

Read it as: "derivative of the outside (leaving the inside alone) times derivative of the inside." The Leibniz form $\frac{dy}{du} \cdot \frac{du}{dx}$ makes the chain visual — the $du$ terms appear to "cancel," leaving $\frac{dy}{dx}$.

How to Apply It — Step by Step

Given $h(x) = \sin(x^2)$:

Identify outer and inner functions

Outer: $f(u) = \sin(u)$ | Inner: $g(x) = x^2$

Differentiate the outer function

$f'(u) = \cos(u)$ — leave the inner untouched: $\cos(x^2)$

Differentiate the inner function

$g'(x) = 2x$

Multiply

$h'(x) = \cos(x^2) \cdot 2x = 2x\cos(x^2)$

Quick Examples

Example 1

$(3x+1)^5$

Outer: $u^5$, inner: $3x+1$
$= 5(3x+1)^4 \cdot 3 = 15(3x+1)^4$

Example 2

$e^{x^2}$

Outer: $e^u$, inner: $x^2$
$= e^{x^2} \cdot 2x = 2xe^{x^2}$

Example 3

$\ln(\cos x)$

Outer: $\ln u$, inner: $\cos x$
$= \frac{1}{\cos x} \cdot (-\sin x) = -\tan x$

Example 4 — Triple chain

$\sin^2(e^x)$

Three layers: $u^2$, $\sin(v)$, $e^x$
$= 2\sin(e^x)\cos(e^x)\cdot e^x$

Common Mistakes

✗ Forgetting to multiply by the inner derivative

Writing $\frac{d}{dx}[e^{x^2}] = e^{x^2}$ is wrong. You must multiply by $\frac{d}{dx}[x^2] = 2x$. The correct answer is $2xe^{x^2}$. This is the single most common chain rule error.

✗ Differentiating the inner function in the outer's position

For $\sin(x^2)$: writing $\cos(2x)$ instead of $\cos(x^2) \cdot 2x$. The outer function's argument stays as the original inner function — only the outer shell gets differentiated first.

✗ Missing a chain on deeply nested functions

Every layer of composition needs its own chain. For $\sin^2(e^x)$, there are three layers — missing any one gives the wrong answer. Count your compositions before you differentiate.

Related Concepts

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