Derivative (Power Rule) Calculator Formula

Understand the math behind the derivative (power rule) calculator. Each variable explained with a worked example.

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Formulas Used

New Coeff

new_coeff = coeff * n

New Exp

new_exp = n - 1

f'(x) at given x

derivative_at_x = coeff * n * pow(x_val, n - 1)

Original At X

original_at_x = coeff * pow(x_val, n)

Variables

VariableDescriptionDefault
coeffCoefficient (a)5
nExponent (n)3
x_valEvaluate at x =2

How It Works

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

    • Worked Example

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

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

    Frequently Asked Questions

    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.

    Learn More

    Guide

    Introduction to Derivatives - Complete Guide

    Learn the fundamentals of derivatives in calculus. Covers the definition, power rule, product rule, chain rule, and practical applications with clear examples.

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