The product rule and quotient rule, explained

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

The product rule differentiates two functions multiplied together, and the quotient rule handles one function divided by another. Together they cover a huge share of the derivatives you will meet in AP Calculus and A level Maths.

Why you need special rules

It is tempting to think the derivative of a product is the product of the derivatives. It is not. Take f(x) = x · x = x². Its derivative is 2x. But multiplying the derivatives of x and x gives 1 × 1 = 1, which is wrong. Products and quotients need their own rules.

The product rule

If y = u · v, where u and v are both functions of x, then

dy/dx = u'v + uv'

In words: derivative of the first times the second, plus the first times the derivative of the second. The order of the two terms does not matter, because they are added.

Where the product rule comes from

Picture a rectangle with sides u and v, so its area is uv. If u grows by a small amount Δu and v grows by Δv, the area grows by a strip vΔu along one side, a strip uΔv along the other, and a tiny corner ΔuΔv. Divide by Δx and let it shrink: the strips give u'v and uv', while the corner disappears because it is the product of two small quantities. That is exactly the product rule.

The quotient rule

If y = u/v, then

dy/dx = (u'v − uv') / v²

A well-known memory aid, with u as “high” and v as “low”: “low d-high minus high d-low, over the square of what's below”. Unlike the product rule, the order matters here: the subtraction must be u'v − uv', or the answer has the wrong sign.

The quotient rule can be derived from the product rule. Write y = u · v⁻¹, then use the product rule and the chain rule: y' = u'v⁻¹ − uv'v⁻² = (u'v − uv')/v².

Worked examples: product rule

Example 1: a polynomial times a trig function

Differentiate y = x² sin x.

• u = x², u' = 2x; v = sin x, v' = cos x.
• dy/dx = 2x sin x + x² cos x = x(2 sin x + x cos x).

Example 2: an exponential

Differentiate y = x³e^x.

• u' = 3x², v' = e^x.
• dy/dx = 3x²e^x + x³e^x = x²e^x(3 + x). Factorising helps when you then need to solve dy/dx = 0.

Example 3: with the chain rule

Differentiate y = x(2x + 1)⁴.

• u = x, u' = 1. v = (2x + 1)⁴, so v' = 4(2x + 1)³ × 2 = 8(2x + 1)³.
• dy/dx = (2x + 1)⁴ + 8x(2x + 1)³ = (2x + 1)³(2x + 1 + 8x) = (2x + 1)³(10x + 1).

Example 4: three factors

Differentiate y = x e^x sin x.

• For three factors, differentiate one at a time and add: (uvw)' = u'vw + uv'w + uvw'.
• dy/dx = e^x sin x + x e^x sin x + x e^x cos x = e^x(sin x + x sin x + x cos x).

Worked examples: quotient rule

Example 5: a rational function

Differentiate y = (x + 1)/(x − 2).

• u = x + 1, u' = 1; v = x − 2, v' = 1.
• dy/dx = (1 · (x − 2) − (x + 1) · 1)/(x − 2)² = −3/(x − 2)².

Example 6: tan x

Use the quotient rule to show that the derivative of tan x is sec²x.

• tan x = sin x/cos x, so u' = cos x and v' = −sin x.
• dy/dx = (cos x · cos x − sin x · (−sin x))/cos²x = (cos²x + sin²x)/cos²x = 1/cos²x = sec²x.

Example 7: an exponential quotient

Differentiate y = e^2x/x.

• u' = 2e^2x, v' = 1.
• dy/dx = (2x e^2x − e^2x)/x² = e^2x(2x − 1)/x².

When to rewrite instead

The quotient rule is not always the fastest route. If the denominator is a single term, split the fraction or use negative powers. For y = (x² + 3)/x, write y = x + 3x⁻¹, so dy/dx = 1 − 3x⁻². If the numerator is a constant, use the chain rule: the derivative of 5/(x² + 1) = 5(x² + 1)⁻¹ is −10x/(x² + 1)². Save the quotient rule for genuine fractions where both top and bottom contain x.

Table questions (AP style)

AP Calculus often gives a table of f, f', g and g' values at certain points and asks for the derivative of a product or quotient at one of them. Substitute the numbers directly into the rule.

Example 8: from a table

f(2) = 3, f'(2) = −1, g(2) = 4, g'(2) = 5. Find h'(2) for h = fg and for k = f/g.

• h'(2) = f'(2)g(2) + f(2)g'(2) = (−1)(4) + (3)(5) = 11.
• k'(2) = (f'(2)g(2) − f(2)g'(2))/g(2)² = (−4 − 15)/16 = −19/16.

Common mistakes

• Multiplying the derivatives: (uv)' is not u'v'.
• Getting the subtraction the wrong way round in the quotient rule.
• Forgetting to square the denominator.
• Forgetting the chain rule inside u or v, such as with (2x + 1)⁴.
• Using the quotient rule when simple rewriting would be quicker and safer.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are the product and quotient rules on AP Calculus AB?

Yes. Both are in Unit 2 of AP Calculus AB and BC, and they are used throughout the rest of the course.

Is the quotient rule in the A level formula booklet?

Most A level Maths formula booklets include the quotient rule. The product rule is usually not given, so you need to learn it.

Key takeaways

• Product rule: (uv)' = u'v + uv'.
• Quotient rule: (u/v)' = (u'v − uv')/v²; order matters.
• Remember: low d-high minus high d-low, over low squared.
• Rewrite simple fractions instead of using the quotient rule.
• Combine with the chain rule whenever u or v is a composite function.

Related guides

• AP Calculus derivative rules
• The chain rule
• Derivatives of exponentials and logarithms