Problem 87E

87. (a) Use the Product Rule twice to prove that if $f, g$, and $h$ are differentiable, then $(f g h)^{\prime}=f^{\prime} g h+f g^{\prime} h+f g h^{\prime}$

(b) Taking $f=g=h$ in part (a), show that

$$\frac{d}{d x}[f(x)]^{3}=3[f(x)]^{2} f^{\prime}(x)$$

(c) Use part (b) to differentiate $y=\left(x^{4}+3 x^{3}+17 x+82\right)^{3}$.

 

Step-by-Step Solution