Activity 5.10 Product and Quotient Rules Together
ΒΆSometimes both the product rule and quotient rule need to be applied when finding a derivative formula.
Subsection Exercises
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.