Chain Rule Worked Examples Cheat Sheet

A printable reference covering the chain rule, composite functions, power, trigonometric, exponential, and logarithmic derivatives for grades 11-12.

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The chain rule is used to differentiate composite functions, where one function is placed inside another. This cheat sheet helps students recognize inside and outside functions and apply the rule step by step. Worked examples are especially useful because chain rule problems often look different even when they follow the same pattern. Students in grades 11-12 need this reference for derivatives involving powers, radicals, trigonometric functions, exponentials, and logarithms. The core idea is to differentiate the outside function while leaving the inside function unchanged, then multiply by the derivative of the inside function. In function notation, if $y = f(g(x))$ , then $\frac{dy}{dx} = f'(g(x))g'(x)$ . In Leibniz notation, if $y = f(u)$ and $u = g(x)$ , then $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$ . The most important skill is identifying the inner function $u$ before differentiating.

Key Facts

The chain rule states that if $y = f(g(x))$ , then $\frac{dy}{dx} = f'(g(x))g'(x)$ .
Using substitution, if $u = g(x)$ and $y = f(u)$ , then $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$ .
For a power of a function, $\frac{d}{dx}[u^n] = n u^{n-1}\frac{du}{dx}$ .
For a radical expression, rewrite first when helpful, such as $\sqrt{u} = u^{\frac{1}{2}}$ , then use $\frac{d}{dx}[u^{\frac{1}{2}}] = \frac{1}{2}u^{-\frac{1}{2}}\frac{du}{dx}$ .
For a sine composite, $\frac{d}{dx}[\sin(u)] = \cos(u)\frac{du}{dx}$ .
For a cosine composite, $\frac{d}{dx}[\cos(u)] = -\sin(u)\frac{du}{dx}$ .
For an exponential composite, $\frac{d}{dx}[e^u] = e^u\frac{du}{dx}$ .
For a logarithmic composite, $\frac{d}{dx}[\ln(u)] = \frac{1}{u}\frac{du}{dx}$ .

Vocabulary

Chain rule: A derivative rule used for composite functions, written as $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$ .
Composite function: A function made by putting one function inside another, such as $f(g(x))$ .
Inner function: The inside expression in a composite function, often labeled $u = g(x)$ .
Outer function: The function applied to the inner expression, such as $f(u)$ when $y = f(g(x))$ .
Derivative: A function that gives the instantaneous rate of change of another function, often written as $\frac{dy}{dx}$ .
Leibniz notation: A notation for derivatives that shows related variables, such as $\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$ .

Common Mistakes to Avoid

Forgetting to multiply by the inner derivative is wrong because the chain rule requires the factor $g'(x)$ in $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$ .
Differentiating the inner function first and stopping is wrong because the outside function must also be differentiated, as in $\frac{d}{dx}[(3x+1)^5] = 5(3x+1)^4\cdot 3$ .
Changing the inside expression while differentiating the outside is wrong because the outside derivative keeps the inner expression unchanged, such as $\frac{d}{dx}[\sin(x^2)] = \cos(x^2)\cdot 2x$ .
Dropping a negative sign in trigonometric derivatives is wrong because $\frac{d}{dx}[\cos(u)] = -\sin(u)\frac{du}{dx}$ .
Treating $\ln(u)$ like $\frac{1}{x}$ is wrong because the derivative is $\frac{1}{u}\frac{du}{dx}$ , not just $\frac{1}{x}$ .

Practice Questions

1 Differentiate $y = (4x^2 - 3x + 1)^6$ .
2 Find $\frac{dy}{dx}$ for $y = \sin(5x^3 - 2x)$ .
3 Differentiate $y = e^{\sqrt{x}}$ .
4 Explain why $\frac{d}{dx}[\ln(2x^2 + 1)]$ needs the chain rule, and identify the inner and outer functions.

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