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MathidiaMathematics Academy
Differentiation
Lesson 32 of 75
Easy practice

The power rule

Objective. Differentiate polynomial and fractional-power expressions.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The power rule states that for f(x) = xⁿ, the derivative is f'(x) = n×xⁿ⁻¹, which also applies to negative and fractional powers. Each term of a polynomial or expression written with powers of x is differentiated separately by multiplying by the original exponent and reducing the exponent by one. For example, if f(x) = 4x³ − 2x + 7, then f'(x) = 12x² − 2, since the derivative of the constant 7 is zero. Fractional powers work identically: the derivative of √x = x^(1/2) is (1/2)x^(−1/2), which equals 1/(2√x).

Worked example

Use the power rule to differentiate f(x) = 2x⁴ − 5x^(1/2) + 6.

1/2
Fraction bar showing 1 out of 2 parts shaded
  1. 1Differentiate 2x⁴ to get 8x³.
  2. 2Differentiate −5x^(1/2) to get −5×(1/2)x^(−1/2) = −(5/2)x^(−1/2).
  3. 3Differentiate the constant 6 to get 0.
  4. 4Combine terms: f'(x) = 8x³ − (5/2)x^(−1/2).

Answer: f'(x) = 8x³ − (5/2)x^(−1/2)

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Common mistake. A common slip is forgetting to reduce the power by one, writing the derivative of x³ as 3x³ instead of 3x². Always both multiply by the old exponent and subtract one from it.

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