冪法則(Power Rule)
\[\frac{d}{dx}x^n=nx^{n-1}\ \ \ \ \ \ n\ne0\]
證明
\[when\ n\ is\ a\ positive\ integer\]
\[\frac{d}{dx}x^n=\lim_{h\rightarrow0}\frac{\left(x+h\right)^n-x^n}{h}=\lim_{h\rightarrow0}\frac{\ x^n+nx^{n-1}h+...-x^n}{h}\]
利用二項式定理(正整數)
\[=\lim_{h\rightarrow0}\frac{n\ x^{n-1}h+...}{h}=\lim_{h\rightarrow0}\left(n\ x^{n-1}+\frac{n\left(n-1\right)}{2}x^{n-2}h+...\right)=n\ x^{n-1}\]
\[when\ n\ is\ a\ negative\ integer\ ,\ let\ m=-n\ \]
\[\frac{d}{dx}x^n=\frac{d}{dx}\frac{1}{x^m}=\frac{-mx^{m-1}}{x^{2m}}=-mx^{-m-1}=nx^{n-1}\]
\[when\ n\ is\ a\ rational\ number,\ let\ say\ n=\frac{p}{q}\ \ \left(p,q\ are\ integer\right)\]
\[let\ y=x^{\frac{p}{q}}\ \ \Rightarrow y^q=x^p\]
\[\frac{d}{dx}y^q=\frac{d}{dx}x^p\]
\[qy^{q-1}\ \frac{dy}{dx}=px^{p-1}\]
\[\frac{dy}{dx}=\frac{x^{p-1}}{y^{q-1}}=\frac{p}{q}\frac{x^{p-1}}{\left(x^{\frac{p}{q}}\right)^{q-1}}=\frac{p}{q}x^{p-1-\frac{p\left(q-1\right)}{q}}=\frac{p}{q}x^{\frac{pq-q-pq+p}{q}}=\frac{p}{q}x^{\frac{p}{q}-1}=nx^{n-1}\]