Calculus I · Unit 2A · lesson

The Product Rule

Concept

Learning objectives

Use and explain the product rule; decide when expansion is preferable.

Products of Changing Functions

Explanation

Before the formulas

The shortcuts in The Product Rule do not replace the limit definition. They package conclusions already justified from it. That distinction matters because a rule applies only when the function has the relevant structure and lies in its domain. A familiar-looking exponent or fraction is not permission to differentiate blindly.

After using a rule, check the result structurally. A derivative of a polynomial should have lower degree. A product derivative should usually contain contributions from both factors. A quotient result should preserve the denominator restriction. These checks are quick enough to use on homework and valuable enough to use on exams.

**Visual study**
When both side lengths of a rectangle change, the area changes because the width changes and because the height changes. The two first-order strips become the two terms in the product rule.

Explanation

Both factors are changing

The tempting but false rule $(fg)'=f'g'$ ignores two first-order contributions. When $x$ changes, the first factor changes while the second still contributes its current size, and the second factor changes while the first contributes its current size. The product rule adds those two effects.

The area model makes this visible: a changing rectangle gains one strip from the width change and another from the height change. The tiny corner where both changes occur is second order and disappears relative to the main strips in the derivative limit.

When two changing factors are multiplied, both can contribute to the change in the product. Revenue equals price times quantity, electrical power equals voltage times current, and the area of a changing rectangle equals length times width. Differentiating only one factor ignores part of the motion.

The product rule is therefore a bookkeeping law for simultaneous change. One term records the effect of changing the first factor while freezing the second; the other records the effect of changing the second while freezing the first.

The derivative of a product is not the product of the derivatives. If both factors change, each contributes to the total change.

Theorem

Product rule

\boxed{\frac{d}{dx}[f(x)g(x)]=f'(x)g(x)+f(x)g'(x)}.

A useful spoken form is: derivative of the first times the second, plus the first times derivative of the second.

Proof idea

The missing term that naive multiplication ignores

Add and subtract $f(x+h)g(x)$ in the product difference quotient. This separates the change into one part caused by $g$ and one part caused by $f$ . After taking limits, the two surviving contributions are $f'g$ and $fg'$ . Geometrically, if a rectangle's width and height both change, the added area contains two main strips; multiplying only the derivatives measures neither strip correctly.

Guided walkthrough

A polynomial times a radical

Differentiate

y=(x^2+1)\sqrt{x}.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

Expand or use the product rule?

For $y=(x+2)(x^2-3x+1)$ , either method works. Expansion produces a polynomial and is likely fastest. For $y=x^2\sin x$ , expansion is impossible, so the product rule is required. Rule choice is part of the mathematics, not clerical decoration.

Common mistake

(fg)'\ne f'g'.

A quick counterexample is $f(x)=g(x)=x$ . The derivative of $x^2$ is $2x$ , while $f'g'=1$ .

Interactive checkproduct-rule-01

Differentiate $x^3\sin x$ .

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Differentiate one factor at a time and add the two contributions.

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Modeling lab

Revenue when price falls as sales rise

A vendor expects to sell $q(t)=200+30t$ units per week after an advertising launch, while the market price falls according to $p(t)=50-2t$ dollars per unit. Revenue is

R(t)=p(t)q(t).

Therefore

R'(t)=p'(t)q(t)+p(t)q'(t).

At $t=4$ , $p=42$ , $q=320$ , $p'=-2$ , and $q'=30$ . Hence

R'(4)=(-2)(320)+(42)(30)=620\text{ dollars/week}^{2}.

The falling price reduces revenue by $640$ dollars per week squared, while growing sales add $1260$ ; the net effect is positive.

Optional advanced note

A first glimpse of differentials

If $f$ changes by $df$ and $g$ changes by $dg$ , then

(f+df)(g+dg)-fg=g\,df+f\,dg+df\,dg.

The last term is second order: it is the product of two small changes. After dividing by the input change and taking a limit, that second-order contribution disappears, leaving the product rule. This "keep the first-order terms" principle becomes central in multivariable calculus and numerical analysis.

After the explanation

Use the section idea

Reading lens

Every rule is a compressed limit calculation; choose the structure before doing algebra.

Mental model

Sums contribute independently, products have two changing contributions, and quotients must account for a changing denominator.

Decision

Name the outermost algebraic structure, then apply the smallest rule set that preserves it.

Common trap

Applying a familiar rule to the wrong outer structure or simplifying after a differentiation error.

Check yourself

Can you justify the primary rule before writing the first derivative symbol?

Interactive checkproduct-extra-01

Differentiate $x^2\sin x$ .

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Show hint

Use two product-rule terms.

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Original instruction with traceable references.

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