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Activity5.12Supplement

Subsection5.12.1Exercises

Apply the constant factor and power rules to find the first derivative of each of the following functions. In each case take the derivative with respect to the variable suggested by the expression on the right side of the equation. Make sure that you assign the proper name to the derivative function. Please note that there is no need to employ either the quotient rule or the product rule to find any of these derivative formulas.

1

\(\fe{f}{x}=\dfrac{\sqrt[11]{x^6}}{6}\)

2

\(y=-\dfrac{5}{t^7}\)

3

\(\fe{y}{u}=12\sqrt[3]{u}\)

4

\(\fe{z}{\alpha}=e\alpha^{\pi}\)

5

\(z=\dfrac{8}{\sqrt{t^7}}\)

6

\(T=\dfrac{4\sqrt[3]{t^7}}{t^2}\)

Apply the constant factor and product rules to find the first derivative of each of the following functions. Write out the Leibniz notation for the product rule as applied to the given function. In each case take the derivative with respect to the variable suggested by the expression on the right side of the equation. Make sure that you assign the proper name to the derivative function.

7

\(y=5\fe{\sin}{x}\fe{\cos}{x}\)

8

\(y=\dfrac{5t^2e^t}{7}\)

9

\(\fe{F}{x}=4x\fe{\ln}{x}\)

10

\(z=x^2\fe{\sin^{-1}}{x}\)

11

\(\fe{T}{t}=(t^2+1)\fe{\tan^{-1}}{t}\)

12

\(T=\dfrac{x^77^x}{3}\)

Apply the constant factor and quotient rules to find the first derivative of each of the following functions. Write out the Leibniz notation for the quotient rule as applied to the given function. In each case take the derivative with respect to the variable suggested by the expression on the right side of the equation. Make sure that you assign the proper name to the derivative function.

13

\(\fe{q}{\theta}=\dfrac{4e^{\theta}}{e^\theta+1}\)

14

\(\fe{u}{x}=2\dfrac{\fe{\ln}{x}}{x^4}\)

15

\(F=\dfrac{\sqrt{t}}{3t^2-5\sqrt{t^3}}\)

16

\(\fe{F}{x}=\dfrac{\fe{\tan}{x}}{\fe{\tan^{-1}}{x}}\)

17

\(p=\dfrac{\fe{\tan^{-1}}{t}}{1+t^2}\)

18

\(y=\dfrac{4}{\fe{\sin}{\beta}-2\fe{\cos}{\beta}}\)

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{align*} \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{align*}
19

Apply the product rule twice to the expression \(\lzoo{x}{\fe{f}{x}\fe{g}{x}\fe{h}{x}}\) to verify the formula stated above for the product of three functions. The first application of the product rule is shown below to help you get you started.

\begin{align*} \lzoo{x}{\fe{f}{x}\fe{g}{x}\fe{h}{x}}&=\lzoo{x}{\fe{f}{x}\cdot\left[\fe{g}{x}\fe{h}{x}\right]} \end{align*}
20

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

Find the first derivative with respect to \(x\) for each of the following functions and for each function find the equation of the tangent line at the stated value of \(x\text{.}\) Make sure that you use appropriate techniques of differentiation.

21

\(\fe{f}{x}=\dfrac{x^3+x^2}{x}\text{;}\) tangent line at \(5\)

22

\(\fe{h}{x}=\dfrac{x}{1+x}\text{;}\) tangent line at \(-2\)

23

\(\fe{K}{x}=\dfrac{1+x}{2x+2}\text{;}\) tangent line at \(8\)

24

\(\fe{r}{x}=3\sqrt[3]{x}e^x\text{;}\) tangent line at \(0\)

25

Find the second derivative of \(y\) with respect to \(t\) if \(y=4\sqrt{t}\,t^5\text{.}\) Make sure that you use appropriate techniques of differentiation and that you use proper notation on both sides of the equal sign.

A function \(f\) is shown in Figure 5.12.1. Each question below is in reference to this function. Please note that the numeric endpoints of each interval answer are all integers.

<<SVG image is unavailable, or your browser cannot render it>>

Figure5.12.1\(y=\fe{f}{x}\)
26

Over what intervals is \(\fd{f}\) positive? Over what intervals is \(\fd{f}\) negative?

27

Over what intervals is \(\fd{f}\) increasing? Over what intervals is \(\fd{f}\) decreasing?

28

Over what intervals is \(\sd{f}\) positive? Over what intervals is \(\sd{f}\) negative?

29

The formula for \(f\) is \(\fe{f}{x}=\frac{x^3}{3}+3x^2-16x\text{.}\) Find the formulas for \(\fd{f}\) and \(\sd{f}\text{,}\) and then graph the derivatives and verify your answers to Exercises 5.12.1.26–29.

A function \(g\) is shown in Figure 5.12.4. Each question below is in reference to this function.

<<SVG image is unavailable, or your browser cannot render it>>

Figure5.12.4\(y=\fe{g}{t}\)
30

Where over the interval \(\cinterval{0}{2\pi}\) does \(\fe{\fd{g}}{t}=0\text{?}\) Please note that the \(t\)-scale in Figure 5.12.4 is \(\frac{\pi}{12}\text{.}\)

31

The formula for \(g\) is \(\fe{g}{t}=3\fe{\sin}{t}+3\fe{\sin^2}{t}-5\text{.}\) Find the formula for \(\fd{g}\) and use that formula to solve the equation \(\fe{\fd{g}}{t}=0\) over the interval \(\cinterval{0}{2\pi}\text{.}\) Compare your answers to those found in Exercise 5.12.1.30.

Hint
32

Find the equation of the tangent line to \(\fd{f}\) at \(\pi\) if \(\fe{f}{x}=\fe{\sin}{2x}\text{.}\)

Hint

In Table 5.12.5 several function and derivative values are given for the functions \(f\) and \(g\text{.}\) The exercises that follow are based upon these two functions.

\(x\) \(\fe{f}{x}\) \(\fe{\fd{f}}{x}\) \(\fe{\sd{f}}{x}\) \(\fe{g}{x}\) \(\fe{\fd{g}}{x}\) \(\fe{\sd{g}}{x}\)
\(0\) \(2\) \(-1\) \(-2\) \(-3\) \(-3\) \(-2\)
\(1\) \(1\) \(0\) \(4\) \(-5\) \(2\) \(16\)
\(2\) \(4\) \(7\) \(10\) \(11\) \(37\) \(58\)
\(3\) \(15\) \(20\) \(16\) \(87\) \(126\) \(124\)
\(4\) \(46\) \(39\) \(22\) \(289\) \(293\) \(214\)
Table5.12.5Various values for derivatives of \(f\) and \(g\)

Suppose that you were asked to find the value of \(\fe{\fd{h}}{2}\) where \(\fe{h}{x}=\fe{f}{x}\fe{g}{x}\text{.}\) The first thing you would need to do is find a formula for \(\fe{\fd{h}}{x}\text{.}\) You would then replace \(x\) with \(2\text{,}\) substitute the appropriate function and derivative values, and simplify. This is illustrated below.

\begin{align*} \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{align*}

Following the process outlined in this example, find each of the following.

33

Find \(\fe{\fd{h}}{4}\text{,}\) where \(\fe{h}{x}=\fe{f}{x}\fe{g}{x}\text{.}\)

34

Find \(\fe{\sd{h}}{2}\text{,}\) where \(\fe{h}{x}=\fe{f}{x}\fe{g}{x}\text{.}\)

35

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

36

Find \(\fe{\fd{p}}{4}\text{,}\) where \(\fe{p}{x}=6\sqrt{x}\fe{f}{x}\text{.}\)

37

Find \(\fe{\fd{r}}{1}\text{,}\) where \(\fe{r}{x}=\left(\fe{g}{x}\right)^2\text{.}\)

38

Find \(\fe{\fd{s}}{2}\text{,}\) where \(\fe{s}{x}=x\fe{f}{x}\fe{g}{x}\text{.}\)

39

Find \(\fe{\fd{F}}{4}\text{,}\) where \(\fe{F}{x}=\sqrt{x}\fe{g}{4}\text{.}\)

40

Find \(\fe{\sd{T}}{0}\text{,}\) where \(\fe{T}{x}=\dfrac{\fe{f}{x}}{e^x}\text{.}\)