Exercise 20

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 formula shown for the product of four functions to determine the derivative with respect to $x$ of the function $p$ defined by $\fe{p}{x}=x^2e^x\fe{\sin}{x}\fe{\cos}{x}$.