The product and quotient rules Cambridge IGCSE Additional Mathematics revision
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In plain words
x² sin x is two functions multiplied together, and you cannot just differentiate each and multiply. There is a rule for a product, and another for one function divided by another.
6 things to know
- Product rule: if y = uv, then dy/dx = u × dv/dx + v × du/dx.
- In words: the first times the derivative of the second, plus the second times the derivative of the first.
- Quotient rule: if y = u/v, then dy/dx = (v × du/dx − u × dv/dx) ÷ v².
- In the quotient rule the order on the top matters, because of the minus sign: the bottom times the derivative of the top comes first.
- Write down u, v and their derivatives before putting anything into the rule.
- Factorise the answer where you can. It is easier to use afterwards, for example to find where the gradient is zero.
Worked example
Differentiate y = xe^(3x).
- u = x, so du/dx = 1. v = e^(3x), so dv/dx = 3e^(3x).
- dy/dx = x × 3e^(3x) + e^(3x) × 1.
- Factorise: dy/dx = e^(3x)(3x + 1).
Worked example
Differentiate y = (5x + 1)/(x − 2).
- u = 5x + 1, so du/dx = 5. v = x − 2, so dv/dx = 1.
- dy/dx = [(x − 2) × 5 − (5x + 1) × 1] ÷ (x − 2)².
- The top is 5x − 10 − 5x − 1 = −11. So dy/dx = −11/(x − 2)².
Tips and tricks
- Product rule: plus. Quotient rule: minus, with the bottom squared underneath.
- Keep brackets round each piece until the end. Most errors are sign errors on the top of a quotient.
It lands in your notebook with its questions as flashcards.
The product and quotient rules: 5 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
What is the product rule for y = uv?
Each function in turn is differentiated while the other is left alone.
What is the derivative of x sin x?
First × derivative of second, plus second × derivative of first.
In the quotient rule for y = u/v, what goes on the bottom?
The denominator is squared.
What is the derivative of x²eˣ?
It can be factorised as xeˣ(x + 2).
What is the derivative of x/(x + 1)?
The top is (x + 1) × 1 − x × 1 = 1.
Quiz
5 questions
Tap an answer and you’ll see straight away whether it’s right, and why.
Worksheet
3 questions, 9 marks. Write your answers on paper, then check them.
The product and quotient rules
Cambridge IGCSE Additional Mathematics 0606 · 9 marks · papermunch.org
Name ______________________________ Date ______________
Differentiate x³ ln x.[3]
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3x² ln x + x². x³ × (1/x) + ln x × 3x².
Differentiate (2x + 1)/(x + 3).[3]
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5/(x + 3)². The top is 2(x + 3) − (2x + 1) = 5.
Differentiate e^(2x) cos x.[3]
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2e^(2x) cos x − e^(2x) sin x.
Answers: The product and quotient rules
- 1. 3x² ln x + x². x³ × (1/x) + ln x × 3x².
- 2. 5/(x + 3)². The top is 2(x + 3) − (2x + 1) = 5.
- 3. 2e^(2x) cos x − e^(2x) sin x.