Power Rule (Derivative)
Derivative of a power function.
Formula
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Example
d/dx(x³) = 3x²
How it works
The power rule tells you how the slope of a simple power function changes: if a variable is raised to a fixed exponent, the derivative is found by bringing the exponent down in front and lowering the exponent by 1. It works because the instantaneous rate of change of x^n follows a consistent pattern, which can be derived from the limit definition of the derivative and the binomial expansion for whole-number exponents. This rule is one of the fastest ways to differentiate polynomials term by term.
When to use it
Use this whenever you need the derivative of an expression like x^n, or each term of a polynomial such as 5x^4 - 2x + 7. It is especially useful in optimization, motion problems, and curve analysis where you need slope or rate of change.
Worked example, step by step
Find d/dx(7x^5) Step 1: Keep the constant 7. Step 2: Apply the power rule to x^5: 5x^4. Step 3: Multiply the results: 7 · 5x^4 = 35x^4. Final answer: d/dx(7x^5) = 35x^4
Common mistakes to avoid
- Forgetting to subtract 1 from the exponent after differentiating
- Changing the constant coefficient instead of leaving it multiplied in front
- Using the power rule on expressions that are not a single power of x without first simplifying
Did You Know?
The power rule is usually the very first derivative students learn — bring the exponent down, subtract one.
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