Product and Quotient Rule Practice Problems with Answers

These product and quotient rule practice problems start with two-factor products, move to quotients, and finish with a triple product, a product inside a quotient and a problem from a table of values. Try each one first. The answers come right after the list, then the full solutions.

The two rules:

For a refresher, see the product rule and quotient rule pages, or the product and quotient rules guide.

Problems

Product rule

1. y = x^2 e^x
2. y = (x^2 + 1)(3x - 2)
3. y = \sqrt{x}\,\ln x
4. y = e^x\cos x

Quotient rule

5. y = \dfrac{x}{x + 1}
6. y = \dfrac{2x - 1}{x^2 + 3}
7. y = \dfrac{\sin x}{x}
8. y = \dfrac{\ln x}{x}
9. y = \dfrac{e^x}{x^2}

Putting them together

10. y = x\,e^x\sin x
11. y = \dfrac{x^2 e^x}{x + 1}
12. Given f(2) = 3, f'(2) = -1, g(2) = 4 and g'(2) = 5, find (fg)'(2) and \left(\dfrac{f}{g}\right)'(2).

Answers

Step-by-step solutions

1. Let f = x^2 and g = e^x, so f' = 2x and g' = e^x:

2. Let f = x^2 + 1 and g = 3x - 2:

You can check by expanding first: y = 3x^3 - 2x^2 + 3x - 2 gives the same derivative.

3. Let f = \sqrt{x} and g = \ln x, so f' = \frac{1}{2\sqrt{x}} and g' = \frac{1}{x}:

Here \frac{\sqrt{x}}{x} = \frac{1}{\sqrt{x}} = \frac{2}{2\sqrt{x}}.

4. The derivative of \cos x is -\sin x:

5. Top x, bottom x + 1, and both derivatives are 1:

6. Top derivative 2, bottom derivative 2x:

7. Top derivative \cos x, bottom derivative 1:

8. Top derivative \frac{1}{x}, bottom derivative 1:

9. Top derivative e^x, bottom derivative 2x:

10. Three factors, so three terms. Differentiate x, then e^x, then \sin x, one at a time:

11. The top is a product. From problem 1, its derivative is (x^2 + 2x)e^x. Now use the quotient rule:

Factor e^x from the top and expand: (x^2 + 2x)(x + 1) - x^2 = x^3 + 2x^2 + 2x = x(x^2 + 2x + 2). So

12. Plug the values into each rule:

Common mistakes on these problems

• Multiplying the derivatives. (x^2e^x)' \ne 2x\cdot e^x. The product rule always gives two terms.
• Subtracting in the wrong order. In problem 6, writing (2x - 1)(2x) - 2(x^2 + 3) on top gives the right size but the wrong sign.
• Forgetting to square the bottom. The quotient rule denominator is g^2, not g.
• Expanding when factoring is easier. In problem 9, factor xe^x out of the top so you can cancel an x.

Keep practicing

These 12 problems use fixed numbers. For unlimited practice with new functions and instant feedback, try the product rule lesson and the quotient rule lesson. When the factors themselves have something inside, like x^2\sin(3x), move on to the chain rule practice problems.