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Activity5.10Product and Quotient Rules Together

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.