---
title: "The Product and Quotient Rules"
description: "Differentiate a product or a quotient of functions whose derivatives are already known, without multiplying it out, and see where each rule comes from in the difference quotient."
canonical: https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv
source: https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv.md
retrieved: 2026-09-12
---

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# The Product and Quotient Rules

Differentiate a product or a quotient of functions whose derivatives are already known, without multiplying it out, and see where each rule comes from in the difference quotient.

Subject: Mathematics · Area: Calculus & Analysis · Ages 18 to 19
Page: https://lightmysky.com/learn/mathematics/the-product-and-quotient-rules-mt_NJ7INJ0jfv

## Ready when they can

- Differentiate x² sin x and simplify the result
- Differentiate a quotient and say which factor belongs on top
- Show by counterexample that the derivative of a product is not the product of the derivatives

## Lesson: Products and quotients, differentiated

The derivative of u times v is u'v + uv'. Each factor takes one turn being differentiated while the other waits unchanged, then you add. Multiplying the derivatives instead is wrong: x times x is x squared with derivative 2x, not 1 times 1.

**Example.** Take x squared times sin x. Call u = x squared and v = sin x, so u' = 2x and v' = cos x. The product rule gives 2x sin x + x squared cos x. The same pattern gives x cos x the derivative cos x - x sin x, since the derivative of cos x is -sin x. At x = pi/2, sin is 1 and cos is 0, so x sin x has gradient 1 there.

The derivative of u over v is (u'v - uv') / v squared. The order on top matters: derivative of the top times the bottom, minus the top times derivative of the bottom. Both rules come from the difference quotient: add and subtract one clever term, split into two limits, and the two pieces appear.

**Tip.** Two habits save marks. Keep the quotient order and the v squared below, since flipped order flips the sign. At a point you only need values: with f(2) = 3, f'(2) = 1, g(2) = 4 and g'(2) = 5, the product gradient is 1 times 4 + 3 times 5 = 19.

**Recap.** Differentiate each factor in turn for a product, and keep the order with the square for a quotient.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [The Epsilon-Delta Definition of a Limit](https://lightmysky.com/learn/mathematics/the-epsilon-delta-definition-of-a-limit-mt_jFzRYQzCmk)
- [The Power Rule for Differentiating Polynomials](https://lightmysky.com/learn/mathematics/the-power-rule-for-differentiating-polynomials-mt_l7f4j44bR3)

## Opens up

- [First-Order Linear Equations and the Integrating Factor](https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8)
- [The Chain Rule](https://lightmysky.com/learn/mathematics/the-chain-rule-mt_kdhl4dmwJn)
- [Partial Derivatives](https://lightmysky.com/learn/mathematics/partial-derivatives-mt_tu11fd9Xk9)
- [Integration by Parts](https://lightmysky.com/learn/mathematics/integration-by-parts-mt_v6jZVs2_tf)