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

 Page 5 / 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the arc length of the graph of f(x) = In (sinx) on the interval [Π/4, Π/2].
Sand falling freely from a lorry form a conical shape whose height is always equal to one-third the radius of the base. a. How fast is the volume increasing when the radius of the base is (1m) and increasing at the rate of 1/4cm/sec Pls help me solve
show that lim f(x) + lim g(x)=m+l
list the basic elementary differentials
Differentiation and integration
yes
Damien
proper definition of derivative
the maximum rate of change of one variable with respect to another variable
terms of an AP is 1/v and the vth term is 1/u show that the sum of uv terms is 1/2(uv+1)
what is calculus?
calculus is math that studies the change in math, such as the rate and distance,
Tamarcus
what are the topics in calculus
Augustine
what is limit of a function?
what is x and how x=9.1 take?
what is f(x)
the function at x
Marc
also known as the y value so I could say y=2x or f(x)= 2x same thing just using functional notation your next question is what is dependent and independent variables. I am Dyslexic but know math and which is which confuses me. but one can vary the x value while y depends on which x you use. also
Marc
up domain and range
Marc
enjoy your work and good luck
Marc
I actually wanted to ask another questions on sets if u dont mind please?
Inembo
I have so many questions on set and I really love dis app I never believed u would reply
Inembo
Hmm go ahead and ask you got me curious too much conversation here
am sorry for disturbing I really want to know math that's why *I want to know the meaning of those symbols in sets* e.g n,U,A', etc pls I want to know it and how to solve its problems
Inembo
and how can i solve a question like dis *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
next questions what do dy mean by (A' n B^c)^c'
Inembo
The sets help you to define the function. The function is like a magic box where you put inside stuff(numbers or sets) and you get out the stuff but in different shapes (forms).
I dont understand what you wanna say by (A' n B^c)^c'
(A' n B (rise to the power of c)) all rise to the power of c
Inembo
Aaaahh
Ok so the set is formed by vectors and not numbers
A vector of length n
But you can make a set out of matrixes as well
I I don't even understand sets I wat to know d meaning of all d symbolsnon sets
Inembo
High-school?
yes
Inembo
am having big problem understanding sets more than other math topics
Inembo
So f:R->R means that the function takes real numbers and provides real numer. For ex. If f(x) =2x this means if you give to your function a real number like 2,it gives you also a real number 2times2=4
pls answer this question *in a group of 40 students, 32 offer maths and 24 offer physics and 4 offer neither maths nor physics , how many offer both maths and physics*
Inembo
If you have f:R^n->R^n you give to your function a vector of length n like (a1,a2,...an) where all a1,.. an are reals and gives you also a vector of length n... I don't know if i answering your question. Otherwise on YouTube you havr many videos where they explain it in a simple way
I would say 24
Offer both
Sorry 20
Actually you have 40 - 4 =36 who offer maths or physics or both.
I know its 20 but how to prove it
Inembo
You have 32+24=56who offer courses
56-36=20 who give both courses... I would say that
solution: In a question involving sets and Venn diagram, the sum of the members of set A + set B - the joint members of both set A and B + the members that are not in sets A or B = the total members of the set. In symbolic form n(A U B) = n(A) + n (B) - n (A and B) + n (A U B)'.
Mckenzie
In the case of sets A and B use the letters m and p to represent the sets and we have: n (M U P) = 40; n (M) = 24; n (P) = 32; n (M and P) = unknown; n (M U P)' = 4
Mckenzie
Now substitute the numerical values for the symbolic representation 40 = 24 + 32 - n(M and P) + 4 Now solve for the unknown using algebra: 40 = 24 + 32+ 4 - n(M and P) 40 = 60 - n(M and P) Add n(M and P), as well, subtract 40 from both sides of the equation to find the answer.
Mckenzie
40 - 40 + n(M and P) = 60 - 40 - n(M and P) + n(M and P) Solution: n(M and P) = 20
Mckenzie
thanks
Inembo
Simpler form: Add the sums of set M, set P and the complement of the union of sets M and P then subtract the number of students from the total.
Mckenzie
n(M and P) = (32 + 24 + 4) - 40 = 60 - 40 = 20
Mckenzie
how do i evaluate integral of x^1/2 In x
first you simplify the given expression, which gives (x^2/2). Then you now integrate the above simplified expression which finally gives( lnx^2).
by using integration product formula
Roha
find derivative f(x)=1/x
-1/x^2, use the chain rule
Andrew
f(x)=x^3-2x
Mul
what is domin in this question
noman
all real numbers . except zero
Roha
please try to guide me how?
Meher
what do u want to ask
Roha
?
Roha
the domain of the function is all real number excluding zero, because the rational function 1/x is a representation of a fractional equation (precisely inverse function). As in elementary mathematics the concept of dividing by zero is nonexistence, so zero will not make the fractional statement
Mckenzie
a function's answer/range should not be in the form of 1/0 and there should be no imaginary no. say square root of any negative no. (-1)^1/2
Roha
domain means everywhere along the x axis. since this function is not discontinuous anywhere along the x axis, then the domain is said to be all values of x.
Andrew
Derivative of a function
Waqar
right andrew ... this function is only discontinuous at 0
Roha
of sorry, I didn't realize he was taking about the function 1/x ...I thought he was referring to the function x^3-2x.
Andrew
yep...it's 1/x...!!!
Roha
true and cannot be apart of the domain that makes up the relation of the graph y = 1/x. The value of the denominator of the rational function can never be zero, because the result of the output value (range value of the graph when x =0) is undefined.
Mckenzie
👍
Roha
Therefore, when x = 0 the image of the rational function does not exist at this domain value, but exist at all other x values (domain) that makes the equation functional, and the graph drawable.
Mckenzie
👍
Roha
Roha are u A Student
Lutf
yes
Roha
What is the first fundermental theory of Calculus?
do u mean fundamental theorem ?
Roha
I want simple integral
for MSc chemistry... simple formulas of integration
aparna
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funny
how are you
funny
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aparna
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aparna
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funny
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funny
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funny
I just formulas
aparna
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aparna
value of log ax cot-x cos-x
aparna
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funny
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funny
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aparna
I am M.sc physics now i am studying in m.phil
funny
so what can i do
aparna
I will send you basic formula for integration after two mint first of all i write then i will send you.
funny
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funny
integration f(X) dx this is basic formula of integration sign is not there you can look integration sign in methematics form... and f(X) my be any function any values
funny
you send me your any ID where i can send you information about integration
funny
funny
Hi
RIZWAN
I don't understand the formula
who's formula
funny
which formula?
Roha
what is the advantages of mathematical economics
Mubarak