# Activity5.10Product and Quotient Rules Together¶ permalink

Sometimes both the product rule and quotient rule need to be applied when finding a derivative formula.

# Subsection5.10.1Exercises

Consider the functions defined by \begin{equation*}\fe{f}{x}=x^2\frac{\fe{\sin}{x}}{e^x}\qquad\fe{g}{x}=\frac{x^2\fe{\sin}{x}}{e^x}\text{.}\end{equation*}

##### 1

Discuss why $f$ and $g$ are in fact two representations of the same function.

##### 2

Find $\fe{\fd{f}}{x}$ by first applying the product rule and then applying the quotient rule (where necessary).

##### 3

Find $\fe{\fd{g}}{x}$ by first applying the quotient rule and then applying the product rule (where necessary).

##### 4

Rigorously establish that the formulas for $\fe{\fd{f}}{x}$ and $\fe{\fd{g}}{x}$ are indeed the same.