The Chain Rule

The Chain Rule

Many functions are composites—one function plugged inside another, like $f(g(x))$. For example, $\sin(x^2)$ combines an outer function $f(u) = \sin(u)$ with an inner function $g(x) = x^2$. The chain rule is the fundamental tool for differentiating such compositions:

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

In plain terms: differentiate the outer function (keeping the inner function unchanged), then multiply by the derivative of the inner function.

Identifying Outer and Inner Functions

The critical first step is recognizing which function is outer and which is inner. Ask yourself: what would I compute last if I plugged in a number? That final operation is the outer function. For $\sqrt{3x+1}$, you first compute $3x+1$, then take the square root—so the square root is outer ($f(u)=\sqrt{u}$) and $3x+1$ is inner ($g(x)=3x+1$).

Quick Examples

Power: For $y = (x^2 + 1)^5$, differentiate the power to get $5(x^2+1)^4$, then multiply by the derivative of the inner function: $\frac{dy}{dx} = 5(x^2+1)^4 \cdot 2x = 10x(x^2+1)^4$

Trigonometric: For $y = \sin(3x)$, the outer derivative is $\cos(3x)$, multiplied by the inner derivative $3$: $\frac{dy}{dx} = 3\cos(3x)$

Multiple Layers

Some composites nest deeper. For $y = \sin(e^{2x})$, apply the chain rule repeatedly from outside to inside: $\frac{dy}{dx} = \cos(e^{2x}) \cdot e^{2x} \cdot 2 = 2e^{2x}\cos(e^{2x})$

Each layer contributes one factor; multiply all factors together.

Common Pitfalls

The most frequent error is omitting the inner derivative entirely—writing $\frac{d}{dx}\sin(3x) = \cos(3x)$ instead of $3\cos(3x)$. Another mistake is misidentifying the outer function when the inner function is itself a sum or product. Always verify: does your answer preserve the original inner function unchanged inside the outer derivative?