# 3.9 Derivatives of exponential and logarithmic functions  (Page 5/6)

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$f\left(x\right)={2}^{4x}+4{x}^{2}$

${2}^{4x+2}·\text{ln}\phantom{\rule{0.1em}{0ex}}2+8x$

$f\left(x\right)={3}^{\text{sin}\phantom{\rule{0.1em}{0ex}}3\text{x}}$

$f\left(x\right)={x}^{\pi }·{\pi }^{x}$

$\pi {x}^{\pi -1}·{\pi }^{x}+{x}^{\pi }·{\pi }^{x}\text{ln}\phantom{\rule{0.1em}{0ex}}\pi$

$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(4{x}^{3}+x\right)$

$f\left(x\right)=\text{ln}\sqrt{5x-7}$

$\frac{5}{2\left(5x-7\right)}$

$f\left(x\right)={x}^{2}\text{ln}\phantom{\rule{0.1em}{0ex}}9x$

$f\left(x\right)=\text{log}\phantom{\rule{0.1em}{0ex}}\left(\text{sec}\phantom{\rule{0.1em}{0ex}}x\right)$

$\frac{\text{tan}\phantom{\rule{0.1em}{0ex}}x}{\text{ln}\phantom{\rule{0.1em}{0ex}}10}$

$f\left(x\right)={\text{log}}_{7}{\left(6{x}^{4}+3\right)}^{5}$

$f\left(x\right)={2}^{x}·{\text{log}}_{3}{7}^{{x}^{2}-4}$

${2}^{x}·\text{ln}\phantom{\rule{0.1em}{0ex}}2·{\text{log}}_{3}{7}^{{x}^{2}-4}+{2}^{x}·\frac{2x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}7}{\text{ln}\phantom{\rule{0.1em}{0ex}}3}$

For the following exercises, use logarithmic differentiation to find $\frac{dy}{dx}.$

$y={x}^{\sqrt{x}}$

$y={\left(\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right)}^{4x}$

${\left(\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right)}^{4x}\left[4·\text{ln}\phantom{\rule{0.1em}{0ex}}\left(\text{sin}\phantom{\rule{0.1em}{0ex}}2x\right)+8x·\text{cot}\phantom{\rule{0.1em}{0ex}}2x\right]$

$y={\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x\right)}^{\text{ln}\phantom{\rule{0.1em}{0ex}}x}$

$y={x}^{{\text{log}}_{2}x}$

${x}^{{\text{log}}_{2}x}·\frac{2\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}x}{x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}2}$

$y={\left({x}^{2}-1\right)}^{\text{ln}\phantom{\rule{0.1em}{0ex}}x}$

$y={x}^{\text{cot}\phantom{\rule{0.1em}{0ex}}x}$

${x}^{\text{cot}\phantom{\rule{0.1em}{0ex}}x}·\left[\text{−}{\text{csc}}^{2}x·\text{ln}\phantom{\rule{0.1em}{0ex}}x+\frac{\text{cot}\phantom{\rule{0.1em}{0ex}}x}{x}\right]$

$y=\frac{x+11}{\sqrt[3]{{x}^{2}-4}}$

$y={x}^{-1\text{/}2}{\left({x}^{2}+3\right)}^{2\text{/}3}{\left(3x-4\right)}^{4}$

${x}^{-1\text{/}2}{\left({x}^{2}+3\right)}^{2\text{/}3}{\left(3x-4\right)}^{4}·\left[\frac{-1}{2x}+\frac{4x}{3\left({x}^{2}+3\right)}+\frac{12}{3x-4}\right]$

[T] Find an equation of the tangent line to the graph of $f\left(x\right)=4x{e}^{\left({x}^{2}-1\right)}$ at the point where

$x=-1.$ Graph both the function and the tangent line.

[T] Find the equation of the line that is normal to the graph of $f\left(x\right)=x·{5}^{x}$ at the point where $x=1.$ Graph both the function and the normal line.

$y=\frac{-1}{5+5\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}5}x+\left(5+\frac{1}{5+5\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}5}\right)$

[T] Find the equation of the tangent line to the graph of ${x}^{3}-x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}y+{y}^{3}=2x+5$ at the point where $x=2.$ ( Hint : Use implicit differentiation to find $\frac{dy}{dx}.\right)$ Graph both the curve and the tangent line.

Consider the function $y={x}^{1\text{/}x}$ for $x>0.$

1. Determine the points on the graph where the tangent line is horizontal.
2. Determine the points on the graph where ${y}^{\prime }>0$ and those where ${y}^{\prime }<0.$

a. $x=e~2.718$ b. $\left(e,\infty \right),\left(0,e\right)$

The formula $I\left(t\right)=\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{{e}^{t}}$ is the formula for a decaying alternating current.

1. Complete the following table with the appropriate values.
$t$ $\frac{\text{sin}\phantom{\rule{0.1em}{0ex}}t}{{e}^{t}}$
0 (i)
$\frac{\pi }{2}$ (ii)
$\pi$ (iii)
$\frac{3\pi }{2}$ (iv)
$2\pi$ (v)
$2\pi$ (vi)
$3\pi$ (vii)
$\frac{7\pi }{2}$ (viii)
$4\pi$ (ix)
2. Using only the values in the table, determine where the tangent line to the graph of $I\left(t\right)$ is horizontal.

[T] The population of Toledo, Ohio, in 2000 was approximately 500,000. Assume the population is increasing at a rate of 5% per year.

1. Write the exponential function that relates the total population as a function of $t.$
2. Use a. to determine the rate at which the population is increasing in $t$ years.
3. Use b. to determine the rate at which the population is increasing in 10 years.

a. $P=500,000{\left(1.05\right)}^{t}$ individuals b. ${P}^{\prime }\left(t\right)=24395·{\left(1.05\right)}^{t}$ individuals per year c. $39,737$ individuals per year

[T] An isotope of the element erbium has a half-life of approximately 12 hours. Initially there are 9 grams of the isotope present.

1. Write the exponential function that relates the amount of substance remaining as a function of $t,$ measured in hours.
2. Use a. to determine the rate at which the substance is decaying in $t$ hours.
3. Use b. to determine the rate of decay at $t=4$ hours.

[T] The number of cases of influenza in New York City from the beginning of 1960 to the beginning of 1961 is modeled by the function

$N\left(t\right)=5.3{e}^{0.093{t}^{2}-0.87t},\left(0\le t\le 4\right),$

where $N\left(t\right)$ gives the number of cases (in thousands) and t is measured in years, with $t=0$ corresponding to the beginning of 1960.

1. Show work that evaluates $N\left(0\right)$ and $N\left(4\right).$ Briefly describe what these values indicate about the disease in New York City.
2. Show work that evaluates ${N}^{\prime }\left(0\right)$ and ${N}^{\prime }\left(3\right).$ Briefly describe what these values indicate about the disease in the United States.

a. At the beginning of 1960 there were 5.3 thousand cases of the disease in New York City. At the beginning of 1963 there were approximately 723 cases of the disease in the United States. b. At the beginning of 1960 the number of cases of the disease was decreasing at rate of $-4.611$ thousand per year; at the beginning of 1963, the number of cases of the disease was decreasing at a rate of $-0.2808$ thousand per year.

[T] The relative rate of change of a differentiable function $y=f\left(x\right)$ is given by $\frac{100·{f}^{\prime }\left(x\right)}{f\left(x\right)}\text{%}.$ One model for population growth is a Gompertz growth function, given by $P\left(x\right)=a{e}^{\text{−}b·{e}^{\text{−}cx}}$ where $a,b,$ and $c$ are constants.

1. Find the relative rate of change formula for the generic Gompertz function.
2. Use a. to find the relative rate of change of a population in $x=20$ months when $a=204,b=0.0198,$ and $c=0.15.$
3. Briefly interpret what the result of b. means.

For the following exercises, use the population of New York City from 1790 to 1860, given in the following table.

New york city population over time
Years since 1790 Population
0 33,131
10 60,515
20 96,373
30 123,706
40 202,300
50 312,710
60 515,547
70 813,669

[T] Using a computer program or a calculator, fit a growth curve to the data of the form $p=a{b}^{t}.$

$p=35741{\left(1.045\right)}^{t}$

[T] Using the exponential best fit for the data, write a table containing the derivatives evaluated at each year.

[T] Using the exponential best fit for the data, write a table containing the second derivatives evaluated at each year.

Years since 1790 $P\text{″}$
0 69.25
10 107.5
20 167.0
30 259.4
40 402.8
50 625.5
60 971.4
70 1508.5

[T] Using the tables of first and second derivatives and the best fit, answer the following questions:

1. Will the model be accurate in predicting the future population of New York City? Why or why not?
2. Estimate the population in 2010. Was the prediction correct from a.?

## Chapter review exercises

True or False ? Justify the answer with a proof or a counterexample.

Every function has a derivative.

False.

A continuous function has a continuous derivative.

A continuous function has a derivative.

False

If a function is differentiable, it is continuous.

Use the limit definition of the derivative to exactly evaluate the derivative.

$f\left(x\right)=\sqrt{x+4}$

$\frac{1}{2\sqrt{x+4}}$

$f\left(x\right)=\frac{3}{x}$

Find the derivatives of the following functions.

$f\left(x\right)=3{x}^{3}-\frac{4}{{x}^{2}}$

$9{x}^{2}+\frac{8}{{x}^{3}}$

$f\left(x\right)={\left(4-{x}^{2}\right)}^{3}$

$f\left(x\right)={e}^{\text{sin}\phantom{\rule{0.1em}{0ex}}x}$

${e}^{\text{sin}\phantom{\rule{0.1em}{0ex}}x}\text{cos}\phantom{\rule{0.1em}{0ex}}x$

$f\left(x\right)=\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x+2\right)$

$f\left(x\right)={x}^{2}\text{cos}\phantom{\rule{0.1em}{0ex}}x+x\phantom{\rule{0.1em}{0ex}}\text{tan}\phantom{\rule{0.1em}{0ex}}\left(x\right)$

$x\phantom{\rule{0.1em}{0ex}}{\text{sec}}^{2}\left(x\right)+2x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(x\right)+\text{tan}\phantom{\rule{0.1em}{0ex}}\left(x\right)-{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)$

$f\left(x\right)=\sqrt{3{x}^{2}+2}$

$f\left(x\right)=\frac{x}{4}\phantom{\rule{0.1em}{0ex}}{\text{sin}}^{-1}\left(x\right)$

$\frac{1}{4}\left(\frac{x}{\sqrt{1-{x}^{2}}}+{\text{sin}}^{-1}\left(x\right)\right)$

${x}^{2}y=\left(y+2\right)+xy\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)$

Find the following derivatives of various orders.

First derivative of $y=x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x$

$\text{cos}\phantom{\rule{0.1em}{0ex}}x·\left(\text{ln}\phantom{\rule{0.1em}{0ex}}x+1\right)-x\phantom{\rule{0.1em}{0ex}}\text{ln}\phantom{\rule{0.1em}{0ex}}\left(x\right)\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x$

Third derivative of $y={\left(3x+2\right)}^{2}$

Second derivative of $y={4}^{x}+{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}\left(x\right)$

${4}^{x}{\left(\text{ln}\phantom{\rule{0.1em}{0ex}}4\right)}^{2}+2\phantom{\rule{0.1em}{0ex}}\text{sin}\phantom{\rule{0.1em}{0ex}}x+4x\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}x-{x}^{2}\text{sin}\phantom{\rule{0.1em}{0ex}}x$

Find the equation of the tangent line to the following equations at the specified point.

$y={\text{cos}}^{-1}\left(x\right)+x$ at $x=0$

$y=x+{e}^{x}-\frac{1}{x}$ at $x=1$

$T=\left(2+e\right)x-2$

Draw the derivative for the following graphs.

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by $w\left(t\right)=1.9+2.9\phantom{\rule{0.1em}{0ex}}\text{cos}\phantom{\rule{0.1em}{0ex}}\left(\frac{\pi }{6}t\right),$ where t is measured in hours after midnight, and the height is measured in feet.

Find and graph the derivative. What is the physical meaning?

Find ${w}^{\prime }\left(3\right).$ What is the physical meaning of this value?

${w}^{\prime }\left(3\right)=-\frac{2.9\pi }{6}.$ At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

The following questions consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

Wind speeds of hurricane katrina
Hours after Midnight, August 26 Wind Speed (mph)
1 45
5 75
11 100
29 115
49 145
58 175
73 155
81 125
85 95
107 35

Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?

Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

$-7.5.$ The wind speed is decreasing at a rate of 7.5 mph/hr

questions solve y=sin x
Solve it for what?
Tim
you have to apply the function arcsin in both sides and you get arcsin y = acrsin (sin x) the the function arcsin and function sin cancel each other so the ecuation becomes arcsin y = x you can also write x= arcsin y
Ioana
what is the question ? what is the answer?
Suman
there is an equation that should be solve for x
Ioana
ok solve it
Suman
are you saying y is of sin(x) y=sin(x)/sin of both sides to solve for x... therefore y/sin =x
Tyron
or solve for sin(x) via the unit circle
Tyron
what is unit circle
Suman
a circle whose radius is 1.
Darnell
the unit circle is covered in pre cal...and or trigonometry. it is the multipcation table of upper level mathematics.
Tyron
what is function?
A set of points in which every x value (domain) corresponds to exactly one y value (range)
Tim
what is lim (x,y)~(0,0) (x/y)
limited of x,y at 0,0 is nt defined
Alswell
But using L'Hopitals rule is x=1 is defined
Alswell
Could U explain better boss?
emmanuel
value of (x/y) as (x,y) tends to (0,0) also whats the value of (x+y)/(x^2+y^2) as (x,y) tends to (0,0)
NIKI
can we apply l hospitals rule for function of two variables
NIKI
why n does not equal -1
Andrew
I agree with Andrew
Bg
f (x) = a is a function. It's a constant function.
proof the formula integration of udv=uv-integration of vdu.?
Find derivative (2x^3+6xy-4y^2)^2
no x=2 is not a function, as there is nothing that's changing.
are you sure sir? please make it sure and reply please. thanks a lot sir I'm grateful.
The
i mean can we replace the roles of x and y and call x=2 as function
The
if x =y and x = 800 what is y
y=800
800
Bg
how do u factor the numerator?
Nonsense, you factor numbers
Antonio
You can factorize the numerator of an expression. What's the problem there? here's an example. f(x)=((x^2)-(y^2))/2 Then numerator is x squared minus y squared. It's factorized as (x+y)(x-y). so the overall function becomes : ((x+y)(x-y))/2
The
The problem is the question, is not a problem where it is, but what it is
Antonio
I think you should first know the basics man: PS
Vishal
Yes, what factorization is
Antonio
Antonio bro is x=2 a function?
The
Yes, and no.... Its a function if for every x, y=2.... If not is a single value constant
Antonio
you could define it as a constant function if you wanted where a function of "y" defines x f(y) = 2 no real use to doing that though
zach
Why y, if domain its usually defined as x, bro, so you creates confusion
Antonio
Its f(x) =y=2 for every x
Antonio
Yes but he said could you put x = 2 as a function you put y = 2 as a function
zach
F(y) in this case is not a function since for every value of y you have not a single point but many ones, so there is not f(y)
Antonio
x = 2 defined as a function of f(y) = 2 says for every y x will equal 2 this silly creates a vertical line and is equivalent to saying x = 2 just in a function notation as the user above asked. you put f(x) = 2 this means for every x y is 2 this creates a horizontal line and is not equivalent
zach
The said x=2 and that 2 is y
Antonio
that 2 is not y, y is a variable 2 is a constant
zach
So 2 is defined as f(x) =2
Antonio
No y its constant =2
Antonio
what variable does that function define
zach
the function f(x) =2 takes every input of x within it's domain and gives 2 if for instance f:x -> y then for every x, y =2 giving a horizontal line this is NOT equivalent to the expression x = 2
zach
Yes true, y=2 its a constant, so a line parallel to y axix as function of y
Antonio
Sorry x=2
Antonio
And you are right, but os not a function of x, its a function of y
Antonio
As function of x is meaningless, is not a finction
Antonio
yeah you mean what I said in my first post, smh
zach
I mean (0xY) +x = 2 so y can be as you want, the result its 2 every time
Antonio
OK you can call this "function" on a set {2}, but its a single value function, a constant
Antonio
well as long as you got there eventually
zach
2x^3+6xy-4y^2)^2 solve this
femi
moe
volume between cone z=√(x^2+y^2) and plane z=2
Fatima
It's an integral easy
Antonio
V=1/3 h π (R^2+r2+ r*R(
Antonio
How do we find the horizontal asymptote of a function using limits?
Easy lim f(x) x-->~ =c
Antonio
solutions for combining functions
what is a function? f(x)
one that is one to one, one that passes the vertical line test
Andrew
It's a law f() that to every point (x) on the Domain gives a single point in the codomain f(x)=y
Antonio
is x=2 a function?
The
restate the problem. and I will look. ty
is x=2 a function?
The