Derivative (power) Calculator
Differentiate x^n using the power rule.
Formula first
Overview
The power rule is a foundational principle in calculus used to compute the derivative of a variable raised to a constant real-number exponent. It establishes that the slope of a power function is determined by multiplying the variable term by its current exponent and then decrementing that exponent by exactly one.
Symbols
Variables
n = Power n, x = Variable x, \frac{dy}{dx} = Derivative value
Apply it well
When To Use
When to use: Apply this rule when differentiating any term in the form xⁿ, where n is a constant value. It is valid for all real numbers, including positive integers, negative integers, and fractional exponents representing roots.
Why it matters: This rule allows for the rapid calculation of rates of change without relying on the tedious limit definition of derivatives. It is essential in physics for deriving acceleration from velocity and in economics for determining marginal costs and revenue.
Avoid these traps
Common Mistakes
- Integrating instead of differentiating.
- Forgetting n=0 for constants.
One free problem
Practice Problem
Calculate the instantaneous rate of change of the function f(x) = x³ at the point where x = 2.
Solve for:
Hint: Apply the power rule nxⁿ⁻¹ by substituting 3 for n and 2 for x.
The full worked solution stays in the interactive walkthrough.
References
Sources
- Stewart, James. Calculus: Early Transcendentals.
- Wikipedia: Power rule
- Stewart, Calculus: Early Transcendentals
- Halliday, Resnick, and Walker, Fundamentals of Physics
- Thomas' Calculus: Early Transcendentals, 14th Edition by George B. Thomas Jr., Maurice D. Weir, and Joel Hass
- AQA A-Level Mathematics — Pure (Differentiation)