--- title: "The Product and Quotient Rules" description: "Apply the product and quotient rules to differentiable functions, keeping both product contributions and the quotient's subtraction order. Use the quotient rule only where its denominator is nonzero." 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: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # The Product and Quotient Rules Apply the product and quotient rules to differentiable functions, keeping both product contributions and the quotient's subtraction order. Use the quotient rule only where its denominator is nonzero. 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 - Apply the product rule to formulas or point data, including solving for a missing rate - Apply the quotient rule, check its nonzero-denominator condition and simplify valid results - Use counterexamples to test claims about products and their derivatives ## Lesson: Products and quotients, differentiated A derivative is a limit of difference quotients. Here we use derivative rules, without epsilon-delta proofs. If u and v are differentiable at the input, (uv)' = u'v+uv'. Differentiate each factor in turn while retaining the other, then add. With angles in radians, (sin x)' = cos x and (cos x)' = -sin x. For x⁴sin x, the derivative is 4x³sin x+x⁴cos x. **Example.** To find a product derivative at a point, form its derivative before evaluating. For (2x+1)(x+4), the derivative is 2(x+4)+(2x+1). At 0 this equals 8+1 = 9. For trigonometric products, substitute the angle only after applying the product rule. Multiplying derivatives is not the product rule. For x² times x, the product is x³ and its derivative is 3x². Multiplying the separate derivatives gives 2x, a different function. A factor's value being zero at one point also does not force the product derivative to be zero there: u'v may remain nonzero. If u and v are differentiable at the input and v is nonzero there, (u/v)' = (u'v-uv')/v². The numerator starts with the derivative of the numerator times the denominator. Subtract the numerator times the denominator's derivative. Naming those four parts avoids reversing the subtraction. For a complete quotient derivative, use both numerator products and square the original denominator. For cos x/(x+2), the result is [-(x+2)sin x-cos x]/(x+2)², valid for x ≠ -2. The derivatives of numerator and denominator alone cannot be divided to obtain the quotient derivative. **Example.** For h(x) = (3x+2)/(x+4), the numerator of h' is 3(x+4)-(3x+2) = 10. Thus h' = 10/(x+4)² for x ≠ -4. At 1, h' = 10/25 = 0.4. Check that the denominator is nonzero at the requested input before evaluating. Point data suffice when the functions are differentiable there. With u(0) = 2, u'(0) = 3, v(0) = 5 and v'(0) = 4, the product derivative is 3(5)+2(4) = 23. The equation can also be solved for a missing derivative. For a quotient, use the same point data in (u'v-uv')/v² and require v ≠ 0. A cosine product needs the negative cosine derivative. For x²cos x, the two contributions are 2x cos x and -x²sin x. Retain the original argument in each trigonometric factor. A quotient that simplifies algebraically may be differentiated in its simplified form only on the original domain. *(drawing: For a quotient, also check that the denominator is nonzero at the input.)* **Recap.** Products add two derivative contributions. Quotients subtract in order and divide by the squared nonzero denominator. *Original product-rule and quotient-rule lesson, questions and flashcards written for this learning dataset.* ## 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)