The Product & Quotient Rules

You’ll be able to

Differentiate products of functions with the product rule
Differentiate quotients with the quotient rule
Choose between rewriting and applying a rule based on the expression

Why you cannot just multiply derivatives

It is tempting to guess that the derivative of a product is the product of the derivatives — but that is false. The correct product rule is d/dx[f·g] = f′·g + f·g′: differentiate the first times the second, plus the first times the derivative of the second. A quick check with f = g = x shows why: (x·x)′ = 2x, not x′·x′ = 1. The cross-terms matter.

The quotient rule

For a quotient, the quotient rule is d/dx[f/g] = (f′·g − f·g′)/g². Memorize the order: "low d-high minus high d-low, over low squared". The subtraction makes order matter — swapping the two terms flips the sign of your whole answer, a very common mistake. The denominator is always the bottom function squared.

Sometimes rewriting beats the rule

Not every quotient needs the quotient rule. If the denominator is a single power, dividing through first is faster: (x³ + x)/x = x² + 1, differentiate to 2x. Reserve the quotient rule for genuine ratios where the denominator cannot be split away cleanly. Choosing the lighter tool saves both time and algebra errors.

Product rule

d/dx[f·g] = f′·g + f·g′

Derivative of first times second, plus first times derivative of second. Not f′·g′.

Quotient rule

d/dx[f/g] = (f′·g − f·g′) / g²

Low d-high minus high d-low, all over low squared. The order of subtraction is essential.

Worked example

Differentiate y = (2x + 1)/(x² + 3) using the quotient rule.

1.Identify f = 2x + 1 (so f′ = 2) and g = x² + 3 (so g′ = 2x).
2.Apply the quotient rule: (f′·g − f·g′)/g² = [2(x² + 3) − (2x + 1)(2x)] / (x² + 3)².
3.Expand the numerator: (2x² + 6) − (4x² + 2x) = 2x² + 6 − 4x² − 2x.
4.Combine like terms: −2x² − 2x + 6.

Answer: y′ = (−2x² − 2x + 6)/(x² + 3)².

Watch out

The quotient rule numerator is f′g − fg′, in that exact order. Because of the minus sign, writing the terms backward flips every sign in your answer. Always put the derivative-of-the-top term first.

Checkpoint

If h(x) = x²·sin(x), what is h′(x)? (Recall d/dx[sin x] = cos x.)

Checkpoint

Using the quotient rule, what is the numerator of d/dx[x/(x + 1)] before simplifying?

On the exam

Before reaching for the quotient rule, ask whether the denominator is a single term you can divide out. Rewriting (x³ + x)/x as x² + 1 turns a quotient-rule problem into a one-line power-rule problem and removes a chance to slip on signs.

Answer the 2 checkpoints as you read.