Power Rule

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Section 2.2 Power Rule

We can generalize the earlier example ($g(x)=x^2$) to arbitrary positive integer powers. Suppose $h(x)=x^n\text{.}$ From the definition, we have

\begin{equation} h(x+dx) \approx \frac{dh}{dx}\,dx\text{,}\tag{2.2.1} \end{equation}

and expanding the left-hand side yields

\begin{equation} (x+dx)^n = x^n + nx^{n-1}\,dx+\frac{n(n-1)}{2}x^{n-2}\,dx^2+...\text{.}\tag{2.2.2} \end{equation}

Ignoring higher powers of $dx$ as before, we obtain

\begin{equation} \frac{dh}{dx} = nx^{n-1}\tag{2.2.3} \end{equation}