The Chain Rule

We started using the Power Rule For Functions in section 2.3. Now we can easily prove it. 

Power Rule For Functions: If $y=f^{n}(x)$ and $f$ is differentiable, then $\frac{d y}{d x}=n \cdot f^{n-1}(x) \cdot f^{\prime}(x)$. 

Proof: $y=f^{n}(x)$ is $y=u^{n}$ with $u=f(x)$. Then $\frac{d y}{d u}=n \cdot u^{n-1}$ and $\frac{d u}{d x}=f^{\prime}(x)$ so by the Chain Rule, $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{dy}}{\mathrm{du}} \cdot \frac{\mathrm{du}}{\mathrm{dx}}=\mathrm{n} \cdot \mathrm{u}^{\mathrm{n}-1} \cdot \mathrm{f}^{\prime}(\mathrm{x})=\mathrm{n} \cdot \mathrm{f}^{\mathrm{n}-1}(\mathrm{x}) \cdot \mathrm{f}^{\prime}(\mathrm{x})$