Lecture 7 · Barnett ch. 2 · about 45 min

Product, Quotient and Chain Rules

Differentiating products, quotients and compositions: the three rules that handle every function built by multiplying, dividing or nesting simpler pieces, with average cost and revenue as the running examples.

By the end you can

Apply the product rule to differentiate a function built by multiplying two others
Apply the quotient rule to differentiate average cost and similar ratios
Apply the chain rule to differentiate a composite function
Combine all three rules on a single multi-layered expression

Why the sum rule is not enough

The sum rule lets you differentiate term by term, but it says nothing about a product like $f(x) = x^2 \cdot e^x$ or revenue $R(x) = p(x) \cdot x$ , where price itself depends on quantity. $(fg)' \neq f'g'$ — multiplying the derivatives separately gives the wrong answer, so a dedicated rule is needed.

The product rule

Product ruleKey formulaFormula sheet →

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

$f, g$: differentiable functions

Use it when two functions of x are multiplied together, and neither is a plain constant.

The quotient rule

Average cost, $\bar C(x) = C(x)/x$ , is a ratio of two functions of $x$ . Differentiating it needs the quotient rule.

Quotient ruleKey formulaFormula sheet →

\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2}

$f$: numerator
$g$: denominator, g(x) \neq 0

Use it when one function is divided by another, and the denominator is not simply a constant.

The chain rule

Many economic quantities are built by nesting one function inside another: cost as a function of labour, labour as a function of time. The chain rule differentiates a composition, exactly the composition idea from the functions lesson.

Chain ruleKey formulaFormula sheet →

\frac{d}{dx}\,f\big(g(x)\big) = f'\big(g(x)\big)\cdot g'(x)

$f$: outer function
$g$: inner function

Use it when one expression is raised to a power, or otherwise wrapped around another expression in x, such as $(3x^2+1)^5$.

Combining the rules

Realistic problems often need more than one rule at once: a product where one factor is itself a composition, for example.

Exam practice

Summary and review

Product rule: $(fg)' = f'g + fg'$ . Never multiply the two derivatives together.
Quotient rule: $\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$ . Order matters in the numerator.
Chain rule: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$ : differentiate outer, keep inner, multiply by inner’s derivative.
Realistic expressions often combine two or three rules in one problem.
Average cost’s derivative, via the quotient rule, tells you whether average cost is rising or falling.