Exercise 35 — Calculus Lab Manual

In this table several function and derivative values are given for the functions $f$ and $g$ . The exercises that follow are based upon these two functions.

Various values for derivatives of $f$ and $g$
$x$	$\fe{f}{x}$	$\fe{\fd{f}}{x}$	$\fe{\sd{f}}{x}$	$\fe{g}{x}$	$\fe{\fd{g}}{x}$	$\fe{\sd{g}}{x}$
$0$	$2$	$-1$	$-2$	$-3$	$-3$	$-2$
$1$	$1$	$0$	$4$	$-5$	$2$	$16$
$2$	$4$	$7$	$10$	$11$	$37$	$58$
$3$	$15$	$20$	$16$	$87$	$126$	$124$
$4$	$46$	$39$	$22$	$289$	$293$	$214$

Suppose that you were asked to find the value of $\fe{\fd{h}}{2}$ where $\fe{h}{x}=\fe{f}{x}\fe{g}{x}$ . The first thing you would need to do is find a formula for $\fe{\fd{h}}{x}$ . You would then replace $x$ with $2$ , substitute the appropriate function and derivative values, and simplify. This is illustrated below. $\begin{aligned} \fe{\fd{h}}{x}&=\lzoo{x}{\fe{f}{x}}\fe{g}{x}+\fe{f}{x}\lzoo{x}{\fe{g}{x}} \\ &=\fe{\fd{f}}{x}\fe{g}{x}+\fe{f}{x}\fe{\fd{g}}{x} \\ \fe{\fd{h}}{2}&=\fe{\fd{f}}{2}\fe{g}{2}+\fe{f}{2}\fe{\fd{g}}{2} \\ &=7(11)+4(37) \\ &=225 \end{aligned}$ Following the process outlined in this example, find each of the following.

Find $\fe{\fd{k}}{3}$ , where $\fe{k}{x}=\dfrac{\fe{g}{x}}{\fe{f}{x}}$ .

Simonds, Calculus Lab Manual, §5.12 Supplement, Exercise 35

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