Calculus Honors · AP-aligned study guide

2.3 · Product, quotient, and chain rules

Recognize the structure of a function before choosing a rule, including products containing a composition.

AP 2.9–2.10; 3.1; 3.5 · Reading: Larson & Edwards §§2.3–2.4

Learning objectives

Select and combine product, quotient, and chain rules.
Identify the inner derivative in a composition.
Check derivative expressions at specified inputs.

A product needs two terms

The derivative of a product is not the product of the derivatives. Each factor contributes a term: keep one unchanged while differentiating the other, then add.

For $\displaystyle x^2e^x$ , the derivative is $\displaystyle 2xe^x+x^2e^x$ . Factor afterward if useful; expanding first is optional.

$(uv)'=u' v+uv'$

A quotient needs a squared denominator

For $\displaystyle u/v$ , with $\displaystyle v\ne0$ , the derivative numerator is denominator times derivative of numerator minus numerator times derivative of denominator. Preserve the order to avoid a sign reversal.

For $\displaystyle (x+1)/(x-1)$ , the numerator of the derivative is $\displaystyle (x-1)-(x+1)=-2$ , giving $\displaystyle -2/(x-1)^2$ . The excluded input remains excluded.

$\left(\frac{u}{v}\right)'=\frac{u' v-uv'}{v^2}$

A composition needs the inner rate

For $\displaystyle f(g(x))$ , first differentiate the outer function while retaining the inner input, then multiply by $\displaystyle g'(x)$ . For $\displaystyle (3x+1)^4$ , this gives $\displaystyle 4(3x+1)^3\cdot3$ .

A product may contain a composition. For $\displaystyle x\sin(x^2)$ , product differentiation gives $\displaystyle \sin(x^2)+x\cos(x^2)\cdot2x$ . Identify the structure at each level rather than choosing only one rule.

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

Worked examples

Product with a composition

Differentiate $\displaystyle x^2\sin(3x)$ .

Use the product rule: $\displaystyle 2x\sin(3x)+x^2\frac{d}{dx}\sin(3x)$ .
The chain rule gives $\displaystyle \frac{d}{dx}\sin(3x)=3\cos(3x)$ .
The derivative is $\displaystyle 2x\sin(3x)+3x^2\cos(3x)$ .

A quotient at one point

Find $\displaystyle f'(0)$ if $\displaystyle f(x)=\frac{e^x}{x+1}$ .

The quotient rule gives $\displaystyle f'(x)=\frac{e^x(x+1)-e^x}{(x+1)^2}$ .
Simplify the numerator to $\displaystyle xe^x$ .
At $\displaystyle x=0$ , the result is $\displaystyle 0$ .

Common mistakes

Differentiating a product by multiplying the two derivatives.
Reversing the subtraction in the quotient rule.
Forgetting the inner derivative.

Try it yourself

No calculator required.

1. Differentiate $\displaystyle f(x)=(2x^2+1)^3$ .

Show worked solution

The outer derivative is $\displaystyle 3(2x^2+1)^2$ and the inner derivative is $\displaystyle 4x$ , giving $\displaystyle f'(x)=12x(2x^2+1)^2$ .

2. Differentiate $\displaystyle g(x)=x^2e^{2x}$ .

Show worked solution

Use product and chain rules: $\displaystyle g'(x)=2xe^{2x}+2x^2e^{2x}$ .

3. Find $\displaystyle h'(2)$ for $\displaystyle h(x)=\frac{x^2}{x+1}$ .

Show worked solution

The quotient rule gives $\displaystyle h'(x)=\frac{2x(x+1)-x^2}{(x+1)^2}$ . Substituting $\displaystyle 2$ gives $\displaystyle \frac{8}{9}$ .

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