The Power Rule
Simple Explanation
To differentiate x raised to a power n, bring the exponent down as a multiplying factor and reduce the exponent by 1: d/dx[x^n] = n·x^(n−1). A constant multiplying a function just carries through unchanged: d/dx[c·f(x)] = c·f'(x).
Why Do We Need It?
This is by far the most-used differentiation shortcut — it turns what would be a lengthy limit calculation into a one-line mechanical rule.
Formula
The Power Rule
d/dx[x^n] = n·x^(n−1), d/dx[c·f(x)] = c·f'(x)
To differentiate a power of x, bring the exponent down as a multiplying factor and reduce the exponent by 1. A constant multiplying a function simply carries through unchanged.
- n
- — any real-number exponent
- c
- — a constant multiplier
When to use it: Whenever differentiating a term of the form x^n, or a constant times a function.
Worked Example
Apply the Power Rule
Differentiate f(x) = x⁵.
Why Does This Work?
Applying the limit definition directly to f(x)=x^n and expanding (x+h)^n using the Binomial Theorem produces x^n + n·x^(n−1)·h + (terms with h², h³, ...); subtracting x^n, dividing by h, and letting h→0 leaves only the n·x^(n−1) term — every higher-order term vanishes because it still has a factor of h left over.
Real-Life Example
Differentiating a power-law position function
An object's position is given by s(t) = t³ (a common power-law motion pattern).
The Power Rule instantly gives the velocity function v(t) = s'(t) = 3t², without needing to redo a limit calculation from scratch.
Practice
Differentiate f(x) = 4x³, then evaluate f'(2).
MediumCommon mistake
Forgetting to reduce the exponent by 1 — writing d/dx[x⁵] = 5x⁵ instead of the correct 5x⁴.
Quick Review
- d/dx[x^n] = n·x^(n−1).
- d/dx[c·f(x)] = c·f'(x) — constants carry straight through.
- Bring the exponent down, then subtract 1 from it.