> [!note] Formula > > The power rule states that for all real number $n$, the [[Derivative|derivative]] of a [[Power Function|power function]] is: > > $ > \frac{d}{dx}x^n = nx^{n-1} > $ > [!info] Proof > > Using the [[Exponential Function#Derivative|derivative of the natural exponential function]], the [[Logarithmic Function#Derivative|derivative of the natural logarithm]], and the [[Math/Calculus/Derivative/Chain Rule|chain rule]], the power rule can be shown for all positive values of x. > > $ > \begin{align*} > &\frac{d}{dx}x^n \\ > &= \frac{d}{dx}e^{n\ln{x}} \\ > &= e^{n\ln{x}} \cdot \frac{d}{dx}n\ln{x} \\ > &= x^{n} \cdot \frac{n}{x} \\ > &= \frac{nx^n}{x} \\ > &= nx^{n - 1} > \end{align*} > $ > > However, since for all even functions, $f(-x) = f(x)$, its derivatives will also work for $\frac{d}{dx}f(-x) = \frac{d}{dx}f(x)$, which proves the rule for all negative values of x. > > For all odd functions, $f(-x) = -f(x) \Leftrightarrow -f(-x) = f(x)$, its derivatives will also work for $\frac{d}{dx}f(-x) = \frac{d}{dx}f(x)$ (applying the chain rule on the left), which proves the rule for all negative values of x. > > Since all power functions are either even, odd, or only defined for positive values, the power rule works for all power functions.