The Chain Rule

Differentiating Composite Functions

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The chain rule is a derivative rule for composite functions, where one function is plugged into another. It matters because many real expressions in science, engineering, and economics are built in layers, such as $\sin(x^2)$ , $(3x + 1)^5$ , or $e^{2x}$ . Instead of differentiating the whole expression at once, the chain rule lets you track how a change in the input moves through each layer. This makes complicated derivatives manageable and systematic.

The main idea is to separate a function into an inside part and an outside part. If $y = f(g(x))$ , then the derivative is $\frac{dy}{dx} = f'(g(x)) \times g'(x)$ , which means differentiate the outside while keeping the inside unchanged, then multiply by the derivative of the inside. This reflects how rates of change combine through a sequence of transformations. The rule is especially useful in implicit differentiation, related rates, and any problem where variables depend on other variables.

Key Facts

Chain rule for composite functions: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$
If $y = (u(x))^n$ , then $\frac{dy}{dx} = n(u(x))^{n-1}u'(x)$
If $y = \sin(u(x))$ , then $\frac{dy}{dx} = \cos(u(x))u'(x)$
If $y = e^{u(x)}$ , then $\frac{dy}{dx} = e^{u(x)}u'(x)$
Leibniz form: if $y = f(u)$ and $u = g(x)$ , then $\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$
Example: $\frac{d}{dx}[(3x + 1)^5] = 5(3x + 1)^4 \times 3 = 15(3x + 1)^4$

Vocabulary

Composite function: A function formed when the output of one function becomes the input of another, such as $f(g(x))$ .
Inside function: The inner expression in a composite function, which is evaluated first, such as $g(x)$ in $f(g(x))$ .
Outside function: The function that acts on the result of the inside function, such as $f$ in $f(g(x))$ .
Derivative: A measure of how fast a function changes with respect to its input.
Intermediate variable: A temporary variable like $u = g(x)$ used to break a composite function into simpler parts.

Common Mistakes to Avoid

Forgetting to multiply by the derivative of the inside function, which gives an incomplete derivative because the inner layer also changes with x.
Differentiating the inside and outside separately and then adding them, which is wrong because the chain rule uses multiplication of rates, not addition.
Changing the inside expression while differentiating the outside, which is wrong because you should first treat the inside as a single unchanged quantity.
Using the chain rule on expressions that are not actually composite functions, which can lead to unnecessary or incorrect steps when a simpler rule applies.

Practice Questions

1 Find $\frac{d}{dx}$ of $y = (2x - 5)^4$ .
2 Find $\frac{d}{dx}$ of $y = \cos(x^3)$ .
3 Explain why the derivative of $\sqrt{5x + 1}$ must include a factor from differentiating $5x + 1$ , and describe what goes wrong if that factor is omitted.