Derivatives: the basics

x^n from the definition

The derivative of xnx^n, namely nxn−1nx^{n-1}, follows directly from the definition by expanding (x+h)n(x+h)^n.

(xn)′=nxn−1(x^n)'=nx^{n-1}

When you expand (x+h)n(x+h)^n, the coefficient of the hh term is nxn−1nx^{n-1}. Form the difference quotient, divide by hh, and let h→0h\to 0: only that term survives.

(x+h)n−xnh=nxn−1+(terms with h)\frac{(x+h)^n-x^n}{h}=nx^{n-1}+(\text{terms with } h)

The power rule is not something to memorize; it is the result of this computation.

Examples

  1. 01Differentiateddxx4\displaystyle \frac{d}{dx} x^{4}AnswerHide
    Answer4x3\displaystyle 4x^{3}
  2. 02Differentiateddxx3\displaystyle \frac{d}{dx} x^{3}AnswerHide
    Answer3x2\displaystyle 3x^{2}
  3. 03Differentiateddxx5\displaystyle \frac{d}{dx} x^{5}AnswerHide
    Answer5x4\displaystyle 5x^{4}
  4. 04Differentiateddxx\displaystyle \frac{d}{dx} xAnswerHide
    Answer1\displaystyle 1
  5. 05Differentiateddxx2\displaystyle \frac{d}{dx} x^{2}AnswerHide
    Answer2x\displaystyle 2x

FAQ

Q1Do I need the full expansion of (x+h)n(x+h)^n?Show answerHide
A
Only up to the h¹ term. Every term with h2h^2 or higher still contains h after dividing by h, so it vanishes as h→0h\to 0.

Practice this pattern in the app, 5 problems a day.

Download on the App Store
Before this
The derivative function f'(x)
This topic
x^n from the definition
Next
Power rule, constant multiples, sums and differences

Published 2026-09-03 · Updated 2026-09-05