Exercise 19

Multiple applications of the product rule are required when finding the derivative formula of the product of three of more functions. After simplification, however, the predictable pattern shown below emerges.$$\begin{aligned} \lzoo{x}{fg}&=\lzoo{x}{f}g+f\lzoo{x}{g} \\ \lzoo{x}{fgh}&=\lzoo{x}{f}gh+f\lzoo{x}{g}h+fg\lzoo{x}{h} \\ \lzoo{x}{fghk}&=\lzoo{x}{f}ghk+f\lzoo{x}{g}hk+fg\lzoo{x}{h}k+fgh\lzoo{x}{k} \end{aligned}$$

Apply the product rule twice to the expression $\lzoo{x}{\fe{f}{x}\fe{g}{x}\fe{h}{x}}$ to verify the formula stated above for the product of three functions. The first application of the product rule is shown below to help you get you started.$$\begin{aligned} \lzoo{x}{\fe{f}{x}\fe{g}{x}\fe{h}{x}}&=\lzoo{x}{\fe{f}{x}\cdot\left[\fe{g}{x}\fe{h}{x}\right]} \end{aligned}$$