Power Rule Practice Problems with Answers and Solutions

These power rule practice problems start with single powers and polynomials, then move to negative and fractional exponents, rewriting before you differentiate, and tangent lines. Try each one first. The answers come right after the list, then the full solutions.

The rule:

You will also use the sum and difference rule and the constant multiple rule on almost every problem. For a refresher, read the power rule page.

Problems

Powers and polynomials

1. y = x^7
2. y = 4x^5
3. y = 3x^4 - 2x^3 + 5x - 9

Negative and fractional exponents

4. y = \dfrac{1}{x^3}
5. y = 6\sqrt{x}
6. y = \sqrt[3]{x^2}
7. y = \dfrac{5}{x^2} - \dfrac{2}{\sqrt{x}}

Rewrite first

8. y = (2x + 1)(x - 3)
9. y = \dfrac{x^4 - 3x^2 + 2}{x^2}
10. y = x^\pi + \pi^2

Tangent lines

11. Find the tangent line to y = x^3 - 2x at x = 2.
12. Find the points where y = x^3 - 3x^2 has a horizontal tangent line.

Answers

Step-by-step solutions

1. Bring the 7 down and subtract 1 from the exponent:

2. The constant 4 just comes along:

3. Differentiate term by term. The derivative of 5x is 5 and the derivative of the constant -9 is 0:

4. Rewrite as y = x^{-3}. Then -3 - 1 = -4:

5. Rewrite as y = 6x^{1/2}:

6. Rewrite as y = x^{2/3}. Then \frac{2}{3} - 1 = -\frac{1}{3}:

7. Rewrite both terms as powers: y = 5x^{-2} - 2x^{-1/2}.

8. Expand first: (2x + 1)(x - 3) = 2x^2 - 5x - 3.

The derivative of a product is not the product of the derivatives. Expanding avoids that trap, or you can use the product rule.

9. The bottom is a single term, so split the fraction:

10. \pi is a constant exponent, so the power rule applies to x^\pi. The term \pi^2 \approx 9.87 is just a number, so its derivative is 0:

11. The point is y(2) = 8 - 4 = 4. The slope is y' = 3x^2 - 2, so y'(2) = 10:

12. A horizontal tangent has slope 0:

The points are (0, 0) and (2, 8 - 12) = (2, -4).

Common mistakes on these problems

• Going the wrong way with negative exponents. In problem 4, x^{-3} becomes -3x^{-4}, not -3x^{-2}. Subtracting 1 always moves left on the number line.
• Differentiating a root without rewriting. Write \sqrt[3]{x^2} as x^{2/3} before you start.
• Differentiating a product factor by factor. In problem 8, (2)(1) = 2 is wrong. Expand or use the product rule.
• Treating \pi^2 like a variable. In problem 10, \pi^2 is a constant, so its derivative is 0, not 2\pi.

Keep practicing

These 12 problems use fixed numbers. For unlimited practice with new numbers every time and instant feedback, try the power rule lesson, then the power rule practice. For more examples and the reasoning behind the rule, read the power rule guide. Next, try the product and quotient rule practice problems.