Power Rule (Derivative)

intermediatecalculusderivatives

Differentiates power functions

The Formula

d/dx[xⁿ] = n·xⁿ⁻¹

Variables & Symbols

n

Any real number

x

Variable

When to Use This Formula

  • Taking derivatives of polynomials
  • Optimization problems
  • Rate of change

Worked Examples

Example 1

Problem:

Find derivative of x³

Solution: 3x²

Step-by-Step:

1

Apply power rule: d/dx[x³]

2

Bring down exponent: 3·x³⁻¹

3

Simplify: 3x²

Common Mistakes to Avoid

  • Forgetting to subtract 1 from exponent
  • Errors with negative exponents

Real-World Applications

1

Physics - velocity and acceleration

2

Economics - marginal cost

3

Engineering optimization

Practice Problems

Problem 1: d/dx[x⁵]

Show Answer

Answer: 5x⁴

Problem 2: d/dx[x⁻²]

Show Answer

Answer: -2x⁻³

Related Formulas

product rule

View Formula →

chain rule

View Formula →

quotient rule

View Formula →

Tags:

calculusderivativepower rule

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