Power Rule
Calculus & Advanced MathThe power rule is the differentiation rule that says 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.
Example
$d/dx[x^5] = 5x^4$. Multiply by $5$, reduce the exponent by $1$. That's it!
Key Insight
The power rule handles every polynomial term in one step, making it the most-used rule in all of differential calculus.
Definition
For any real number $n$, $d/dx[x^n] = nx^{n-1}$. It applies to positive integers, fractions, and negative exponents alike. For constants, $n = 0$ and $d/dx[c] = 0$.
Example
$d/dx[x^{1/2}] = (1/2)x^{-1/2} = 1/(2\sqrt{x})$. Also: $d/dx[1/x] = d/dx[x^{-1}] = -x^{-2} = -1/x^2$.
Key Insight
The power rule works for ALL real exponents, not just integers. Combined with the constant-multiple and sum rules, it differentiates any polynomial or rational power in seconds.
Definition
For integer $n$, the power 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. Alternatively, it emerges from the definition of the derivative via limits.
Example
Proof for integer $n$: $(x+h)^n = x^n + nx^{n-1}h + O(h^2)$, so $[(x+h)^n - x^n]/h \to nx^{n-1}$ as $h \to 0$.
Key Insight
The power rule in complex analysis extends to $d/dz[z^n] = nz^{n-1}$ for holomorphic functions. In abstract algebra, it motivates the definition of formal derivatives on polynomial rings.