Chapter 1.7, Problem 95E

The Extended Power Rule (for positive integer powers can be verified using the Product Rule. For example, If $y=\left[f\left(x\right)\right]$ then the Product Rule is applied by recognizing that ${\left[f\left(x\right)\right]}^2=\left[f\left(x\right)\right]\cdot \left[f\left(x\right)\right]$. Therefore,

$\frac{d}{dx}\left(\left[f\left(x\right)\right]\cdot \left[f\left(x\right)\right]\right)=f\left(x\right)\cdot f\text{'}\left(x\right)+f\text{'}\left(x\right)\cdot f\left(x\right)$

$=2f\left(x\right)\cdot f\text{'}\left(x\right)$.

a. Use the Product Rule to show that $\frac{d}{dx}{\left[f\left(x\right)\right]}^3=3{\left[f\left(x\right)\right]}^2\cdot f\text{'}\left(x\right)$.

b. Use the Product Rule to show that $\frac{d}{dx}{\left[f\left(x\right)\right]}^4=4{\left[f\left(x\right)\right]}^3\cdot f\text{'}\left(x\right)$.