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Power Rule for Derivatives

Master the most fundamental derivative rule with step-by-step examples and practice problems.

The Power Rule Formula

If f(x) = x^n, then f'(x) = nx^(n-1)

Where n is any real number

✅ What it means:

  • • Bring down the exponent as a coefficient
  • • Subtract 1 from the original exponent
  • • Works for any real number n

🎯 When to use:

  • • Any function with x raised to a power
  • • Polynomials (x², x³, etc.)
  • • Fractional exponents (√x, ∛x)
  • • Negative exponents (1/x, 1/x²)

Step-by-Step Examples

Example 1: Basic Power

Find: d/dx[x⁵]
Step 1: Identify n = 5
Step 2: Apply power rule: nx^(n-1)
Step 3: 5x^(5-1) = 5x⁴
Answer: 5x⁴

Example 2: Square Root

Find: d/dx[√x]
Step 1: Rewrite as x^(1/2)
Step 2: Apply power rule with n = 1/2
Step 3: (1/2)x^(1/2-1) = (1/2)x^(-1/2)
Step 4: Simplify: 1/(2√x)
Answer: 1/(2√x)

Example 3: Negative Exponent

Find: d/dx[1/x²]
Step 1: Rewrite as x^(-2)
Step 2: Apply power rule with n = -2
Step 3: (-2)x^(-2-1) = -2x^(-3)
Step 4: Simplify: -2/x³
Answer: -2/x³

Practice Problems

Try These:

  1. d/dx[x³]
  2. d/dx[x⁻¹]
  3. d/dx[∛x]
  4. d/dx[x^(2/3)]

Answers:

  1. 3x²
  2. -1/x²
  3. 1/(3∛x²)
  4. (2/3)x^(-1/3)

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Related Derivative Rules

Product Rule

For products of functions

Chain Rule

For composite functions

All Rules

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