Power Rule
The power rule is the differentiation rule that says $\dfrac{d}{dx}[x^n] = nx^{n-1}$, making it easy to differentiate polynomial terms.
Formula
\frac{d}{dx}[x^n] = nx^{n-1}
Definition
The power rule is a quick shortcut: to differentiate $x$ raised to a power, multiply by that power and then subtract $1$ from the exponent. For any real number $n$, $d/dx[x^n] = nx^{n-1}$; it applies to positive integers, fractions, and negative exponents alike, and for constants $n = 0$ gives $d/dx[c] = 0$. For integer $n$, the rule follows from the binomial theorem expansion of $(x+h)^n$; for real $n$, it follows from the identity $x^n = e^{n \ln x}$ and the chain rule.
Example
$d/dx[x^5] = 5x^4$: multiply by $5$, reduce the exponent by $1$. The rule works for fractional and negative exponents too: $d/dx[x^{1/2}] = (1/2)x^{-1/2} = 1/(2\sqrt{x})$, and $d/dx[1/x] = d/dx[x^{-1}] = -x^{-2} = -1/x^2$. The proof for integer $n$ comes from $(x+h)^n = x^n + nx^{n-1}h + O(h^2)$, so $[(x+h)^n - x^n]/h \to nx^{n-1}$ as $h \to 0$.
Key Insight
The power rule handles every polynomial term in one step, making it the most-used rule in all of differential calculus, and combined with the constant-multiple and sum rules it differentiates any polynomial or rational power in seconds. In complex analysis it extends to $d/dz[z^n] = nz^{n-1}$ for holomorphic functions, and in abstract algebra it motivates the definition of formal derivatives on polynomial rings.