What is the power rule?

The power rule states that the derivative of x^n is n×x^(n-1). It works for any real exponent n, including negative and fractional values.

What is the derivative of a constant?

The derivative of any constant is 0. This follows from the power rule since a constant can be written as ax⁰, and 0 × ax^(-1) = 0.

Does the power rule work for negative exponents?

Yes. For example, d/dx [x^(-2)] = -2x^(-3) = -2/x³.

Derivative (Power Rule) Calculator

Calculate the derivative of a term ax^n using the power rule. Shows the resulting coefficient and exponent.

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Coefficient (a)

Exponent (n)

Evaluate at x =

f'(x) at given x

60.0000

New Coeff15.0000

New Exp2

Original At X40.0000

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Power Rule for Derivatives

Rule

d/dx [ax^n] = a × n × x^(n-1)

The power rule is the most fundamental differentiation rule: 1. Bring the exponent down as a multiplier 2. Reduce the exponent by 1

Examples

d/dx [5x³] = 15x²

d/dx [x²] = 2x

d/dx [7x] = 7

d/dx [constant] = 0

Example Calculation

Find the derivative of 5x³ and evaluate at x = 2.

01d/dx [5x³] = 5 × 3 × x^(3-1) = 15x²
02f'(2) = 15 × 2² = 15 × 4 = 60
03f(2) = 5 × 2³ = 5 × 8 = 40

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Derivative (Power Rule) Calculator

Power Rule for Derivatives

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