EZ Math

Calculus · Derivatives

How to Use the Power Rule

Problem

f(x)=x5f(x) = x^5

Answer

f′(x)=5x4f^{\prime}(x) = 5x^4

The power rule is the fastest derivative rule. It works for any power of x, including whole numbers and fractions.

Step-by-step solution

  1. Write the power rule.

    ddx(xn)=nxn−1\frac{d}{dx}\left(x^n\right) = n x^{n-1}

    Why: The rule copies the power down and reduces it by one.

  2. Apply the rule with n = 5.

    f′(x)=5x5−1f^{\prime}(x) = 5x^{5-1}

    Why: The exponent 5 moves in front and the new exponent is 4.

  3. Simplify the exponent.

    f′(x)=5x4f^{\prime}(x) = 5x^4

    Why: The derivative is a new function of x.

Common mistakes

  • Keeping the original power: 5x^5 instead of 5x^4.
  • Forgetting the coefficient: x^4 instead of 5x^4.
  • Applying the rule to a constant term. The derivative of a constant is 0.

Practice problems

Use the same method. Work each problem on paper, then open the answer to check.

  1. f(x)=x3f(x) = x^3

    Show answer

    f′(x)=3x2f^{\prime}(x) = 3x^2

  2. f(x)=x7f(x) = x^7

    Show answer

    f′(x)=7x6f^{\prime}(x) = 7x^6

  3. f(x)=x2f(x) = x^2

    Show answer

    f′(x)=2xf^{\prime}(x) = 2x

  4. f(x)=xf(x) = x

    Show answer

    f′(x)=1f^{\prime}(x) = 1

  5. f(x)=x10f(x) = x^{10}

    Show answer

    f′(x)=10x9f^{\prime}(x) = 10x^9