2.9 The Quotient Rule

If q(x) = f(x)/g(x) and g(x) ≠ 0, then q′(x) = [f′(x)g(x) − f(x)g′(x)] / [g(x)]².

Memory phrase: “low d-high minus high d-low, over low squared,” where high is the top and low is the bottom. The derivative of the top goes first.

Before using it, label the four pieces: the top, its derivative, the bottom and its derivative. Writing them down first prevents most sign and order errors.

In the product rule the two terms are added, so order doesn't matter. In the quotient rule they are subtracted. Swapping them flips the sign of your answer. If you get a slope that should be positive but comes out negative, check the order.

You can build the quotient rule from the product rule. If q = f/g, then f = q·g. Differentiate both sides with the product rule: f′ = q′g + qg′. Solve for q′: q′ = (f′ − qg′)/g. Replace q with f/g and multiply the top and bottom by g: q′ = (f′g − fg′)/g².

So if you ever forget the order, you can rebuild it in a few lines, or rewrite a quotient as f·g⁻¹ and use the product rule with the chain rule (Unit 3).

With table values, plug in carefully: q′(a) = [f′(a)g(a) − f(a)g′(a)] / [g(a)]². Square the bottom value, including its sign: if g(a) = −1, then [g(a)]² = 1.

Rewriting first is often faster and safer than the full rule:

• Constant on the bottom: (x³ + 2)/5 = (1/5)(x³ + 2), so the derivative is 3x²/5.
• Constant on top: 7/x⁴ = 7x⁻⁴, so the derivative is −28x⁻⁵.
• Single-term bottom: (x² + 3)/x = x + 3x⁻¹, so the derivative is 1 − 3/x².
• Otherwise, like (2x + 1)/(x² + 3) or eˣ/x, use the quotient rule (or rewrite as a product with a negative power and use the product and chain rules).

Expand the top, combine like terms and factor if you'll need to set the derivative equal to 0. Usually leave the bottom squared, not expanded: it shows that the bottom is never negative, which helps with sign analysis later (5.3).