Product and Quotient Rules
Rules for differentiating products and ratios.
About this topic
The product rule says (fg)′ = f′g + fg′: derivative of the first times the second plus the first times the derivative of the second. The quotient rule says (f/g)′ = (f′g − fg′)/g². You cannot just multiply or divide derivatives.
Pay attention to the order in the quotient rule, where the numerator is low d-high minus high d-low in the usual chant, since subtraction is not commutative. Sometimes you can avoid the quotient rule by rewriting as a product with a negative exponent.
Example: the derivative of x² sin x is 2x sin x + x² cos x. The derivative of x/(x + 1) is ((1)(x + 1) − x(1))/(x + 1)² = 1/(x + 1)². Checking that a simple result emerges is a good sign that the algebra is right.
These rules are needed whenever two changing quantities multiply or divide, such as revenue equal to price times quantity when both vary with time, or the concentration of a drug equal to amount over volume. The most common mistake is writing (fg)′ = f′g′, which is wrong; test it with f = g = x, where the product is x² with derivative 2x, not 1.
Key ideas
- (uv)′ = u′v + uv′
- (u/v)′ = (u′v − uv′)/v²
- Do not multiply the derivatives
Quick check
Use cases: where Product and Quotient Rules helps
- Homework: review the key idea behind Product and Quotient Rules before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Product and Quotient Rules without notes.
- Visual learning: play with the interactive lab to see how Product and Quotient Rules changes when you adjust its inputs.
- Teaching a friend: explain Product and Quotient Rules in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Product and Quotient Rules?
Rules for differentiating products and ratios.
What will I learn in Product and Quotient Rules?
(uv)′ = u′v + uv′. (u/v)′ = (u′v − uv′)/v². Do not multiply the derivatives.
Is Product and Quotient Rules part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Product and Quotient Rules lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Product and Quotient Rules?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.