Answers
If you're like me and don't remember hyperbolic identities (especially involving inverse functions) off the top of your head, recall the definitions of the hyperbolic cosine and sine:
[tex]\cosh(x) = \dfrac{e^x + e^{-x}}2 \\\\ \sinh(x) = \dfrac{e^x - e^{-x}}2[/tex]
Then differentiating yields
[tex]\dfrac{\mathrm d(\cosh(x))}{\mathrm dx} = \dfrac{e^x-e^{-x}}2 = \sinh(x) \\\\ \dfrac{\mathrm d(\sinh(x))}{\mathrm dx} = \dfrac{e^x+e^{-x}}2 = \cosh(x)[/tex]
so that by the chain rule, if
[tex]y = \cosh\left(4\tanh^{-1}(x)\right)[/tex]
then
[tex]\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\mathrm d\left(\cosh\left(4\tanh^{-1}(x)\right)\right)}{\mathrm dx} = \sinh\left(4\tanh^{-1}(x)\right) \dfrac{\mathrm d\left(4\tanh^{-1}(x)\right)}{\mathrm dx}[/tex]
Now, let [tex]z=4\tanh^{-1}(x)[/tex], so that (•) [tex]\tanh\left(\frac z4\right) = x[/tex].
Recall that
[tex]\tanh(x) = \dfrac{\sinh(x)}{\cosh(x)} = \dfrac{e^x-e^{-x}}{e^x+e^{-x}}[/tex]
and so the derivative of tanh(x) is
[tex]\dfrac{\mathrm d(\tanh(x))}{\mathrm dx} = \dfrac{(e^x+e^{-x})(e^x+e^{-x})-(e^x-e^{-x})(e^x-e^{-x})}{(e^x+e^{-x})^2} \\\\ = \dfrac{4\cosh^2(x)-4\sinh^2(x)}{4\cosh^2(x)} \\\\ = 1-\tanh^2(x) = \mathrm{sech}^2(x)[/tex]
where the last equality follows from the hyperbolic Pythagorean identity,
[tex]\cosh^2(x)-\sinh^2(x) = 1 \implies 1-\tanh^2(x) = \mathrm{sech}^2(x)[/tex]
Differentiating both sides of (•) implicitly with respect to x gives
[tex]\dfrac{\mathrm d\left(\tanh\left(\frac z4\right)\right)}{\mathrm dx} = \dfrac{\mathrm d(x)}{\mathrm dx} \\\\ \mathrm{sech}^2\left(\dfrac z4\right)\dfrac{\mathrm d\left(\frac z4\right)}{\mathrm dx} = 1 \\\\ \dfrac14 \mathrm{sech}^2\left(\dfrac z4\right) \dfrac{\mathrm dz}{\mathrm dx} = 1 \\\\ \dfrac{\mathrm dz}{\mathrm dx} = 4\cosh^2\left(\dfrac z4\right) \\\\ \dfrac{\mathrm dz}{\mathrm dx} = 4\cosh^2\left(\tanh^{-1}(x)\right)[/tex]
So, the derivative we want is the somewhat messy expression
[tex]\boxed{\dfrac{\mathrm dy}{\mathrm dx} = 4\sinh\left(4\tanh^{-1}(x)\right) \cosh^2\left(\tanh^{-1}(x)\right)}[/tex]
and while this could be simplified into a rational expression of x, I would argue for leaving the solution in this form considering how y is given in this form from the start.
In case you are interested, we have
[tex]\cosh\left(\tanh^{-1}(x)\right) = \dfrac1{\sqrt{1-x^2}}[/tex]
and you can instead work on differentiating that; you would end up with
[tex]\dfrac{\mathrm dy}{\mathrm dx} = -\dfrac{16x^3+16x}{(x^2-1)^3}[/tex]
Related Questions
A plane descends 1000 ft while flying
2.7 mi. What is the angle of descent?
Answers
Answer:
4.02°
Step-by-step explanation:
The angle of descent can be obtained using trigonometry ;
Since we have a right angled triangle ;
Let's use the relation:
Sin θ = opposite / hypotenus
Hypotenus = 2.7 miles
1 mile = 5280 feets
2.7 miles = (2.7 * 5280) = 14256 feets
Sin θ = 1000 / 14256
Sin θ = 0.0701459
θ = sin^-1(0.0701459)
θ = 4.0223674
3 that aren’t answered please, ill give brainiest
Answers
Answer:
7: 113.1 yd
In terms of Pi: 36π
8: 326.85 in
In terms of Pi: 104π
6: 452.39 km
In terms of Pi: 144π
Step-by-step explanation:
A circular patio has a radius of 4 yards. The patio is expanded so that the
radius increases by 2 yards. By how many square yards does the area of the
patio increase?
Answers
Answer:
20pi or 62.83
Step-by-step explanation:
when r=4
pi r^2
16pi
when r=6
pi r^2
36pi
36pi-16pi=20pi or 62.83
Let u = <-5, 2>, v = <-2, 6>. Find 4u + 3v.
Answers
Answer:<-26, 26>
Step-by-step explanation: <4(-5), 4(2)> and
<3(-2), 3(6)>
<-20, 8> and <-6, 18>
Now add <(-20+(-6)), (8+18)>
<-26, 26>
Please Help! What decimal is less than 0.4?
Answers
The decimal 0.7 is less than 0.4.
We have to determine
decimal is less than 0.4
What is the decimal?
A decimal number can be defined as a number whose whole number part and the fractional part is separated by a decimal point.
We know that the decimal numbers increase in size as we move to
The right on the number line
0.4 is less than 0.7.
Let us now compare 1.57 and 1.6.
Therefore, The decimal 0.7 is less than 0.4.
To learn more about the decimal visit:
https://brainly.com/question/52825
You want to purchase a new car in 3 years and expect the car to cost $30,000. Your bank offers a savings plan with a guaranteed APR of 5.5% if you make regular monthly deposits. How much should you deposit each month to end up with $30,000 in 3 years?
Answers
Answer:
i think its appr. $1,818
Step-by-step explanation:
im not really sure tho
A video game system costs $350. The sales tax rate is 796, what
is the final cost
Your answer
Answers
Answer:
$377.86
Step-by-step explanation:
Given: Video game system costs $350, sales tax rate is 796.
To find: Final cost
Solve: Calculating Total Cost. Multiply the cost of an item or service by the sales tax to find out the total cost.
3.5 × 7.96 = 27.86
Add the total sales tax to the Item or service cost to get your total cost.
27.86 + 350 = 377.86
The final cost is $377.86
abus
Find the slope of the line shown in the graph below.
5
nts
4
2
1
-6-5-4
2
3
4
5
6
2
Answers
here's the question
Answers
I need help someone help me out!
Answers
[tex]{ \large{ \sf{r {}^{2} = \dfrac{V}{πh} }}}[/tex]
[tex]{ \large{ \sf{ \therefore{r = \sqrt{ \dfrac{V}{πh} } }}}}[/tex]
Option = A is right4. Vivian uses a debit card to pay for a purchase, but she spends more than she has in her bank account. What will happen next?
Answers
Answer:
Vivian will have to get a loan, and pay back the money she spent with her debit card. If she cannot pay back this loan in time, the interest will grow higher and higher. Using a debit card is NOT good because if you don't have the money for something and you buy it anyway, you'll have to pay back double the money if you miss the deadline for the loan.
An engineer traveled 125 mi by car and then an additional 675 mi by plane. The rate of the plane was three times the rate of the car. The total trip took 7 h. Find the rate of the car.
Answers
Answer:
The speed of the car is 50 mph.
The speed of the plane is 150 mph.
Driving time = 185/50 = 3.7 hours
Flying time = 645/150 = 4.3 hours
Step-by-step explanation:
R be the speed of the car.
t is the driving time.
185 = R* t
645 = (3R)*(8-t) <--- the plane is 3 times as fast; the total time is 8 hours, so 8-t hours is spent flying
645 = 24R - 3rt <--- distributive
645 = 24R - 3(185) <--- substitutes rt = 185
645 = 24R - 555
1200 = 24R
R = 50
In an examination, 200 students out of 500, obtained first division. What percentage of student got first division
Answers
Answer: 40%
Step-by-step explanation:
200 out of 500
200/500 or 2/5
1/5 is the same as 20%
Times by 2 you get 40%
2/5 = 40%
Which expression is equivalent to 5c + 3c?
Answers
the correct Number is 8c
rectangle has area 80 square centimeters. the rectangle is dilated using A as a center and a scale factor of 1/2 what is the area of the image
Answers
9514 1404 393
Answer:
20 square centimeters
Step-by-step explanation:
The area of the dilated figure is the original area multiplied by the square of the dilation factor. The center of dilation is irrelevant.
dilated area = (1/2)² × original area
dilated area = 1/4 × 80 cm²
dilated area = 20 cm²
Mr. Wilson invested money in two accounts. His total investment was $20,000. If one account pays 3% in interest and the other pays 6% in interest, how much does he have in each account if he earned a total of $960 in interest in 1 year?
He invested $__ in the 3% account and $__ in the 6% account.
Answers
Answer:
8000 and 12,000
Step-by-step explanation:
Let the amount of money deposited in the 3% account be known as "x"
The amount of money deposited in the 6% account would be "20,000 - x" (aka the remaining amount of money not put into the first account)
3% Interest -> 0.03 *
6% Interest -> 0.06 *
0.03 * x + 0.06 * (20,000 - x) = 960
0.03x + 1200 - 0.06x = 960
-0.03x = -240
x = 8000
20,000 - x
20,000 - 8000 = 12000
The total bill for drinks and a pizza for three people is $1490 before tax the group wants to leave a 15% tip on the cost for the food, not including tax, what will be the amount of the tip?
Answers
Answer:
223.50
Step-by-step explanation:
1490x .15 = 223.50
Section 1 - Question 3 Which expression would produce the greatest product? A 10 × 1/2 10 × 1/ 10 × 100 10 × 3 B C
Answers
Answer:
A: 10 x 1/2 = 15
Step-by-step explanation:
dfghj
what is the y-intercept of the line that goes through (0, 0) (6, 3)
Answers
Answer:
QUESTION:
what is the y-intercept of the line that goes through (0, 0) (6, 3)
ANSWER:
Use the slope formula and the slope-intercept form to find the slope m and y-intercept b .
Slope: 1/2
y-intercept: (0,0)
Step-by-step explanation:
Hope that this helps you out! :)
If you have any questions please put them in the comment section below this answer.
Have a great rest of your day/night!
Please thank me on my profile if this answer has helped you.
For his long distance phone service, Tom pays an 8 monthly fee plus 6 cents per minute. Last month, Tom's long distance bill was $16.34. For how many minutes was Tom billed?
Answers
8 + 0.06x = 16.34
0.06x = 8.34
x = 139
Please lmk if you have any questions
find the surface area of the figure below
Answers
a line is parallel to y=3x-2 and intersects the point (3,5)
Answers
The equation of the line that is parallel to y = 3x - 2 and intersects the point (3, 5) will be y = 3x - 4.
What is the equation of a parallel line?Let the equation of the line be ax + by + c = 0. Then the equation of the parallel line that is parallel to the line ax + by + c = 0 is given as ax + by + d = 0.
The linear equation is given below.
y = 3x - 2
The equation of the line that is parallel to the line y = 3x - 2 will be given as,
y = 3x + c
The equation passes through (3, 5), then we have
5 = 3(3) + c
5 = 9 + c
c = - 4
Then the equation of the line will be y = 3x - 4.
More about the equation of a parallel line link is given below.
https://brainly.com/question/402319
#SPJ2
A rectangular prism has a length of 12 inches, a height of 16 inches, and a with of 20 inches. What's the volume, in cubic inches
Answers
Answer:
3840in²
Step-by-step explanation:
Formula:
V = whl
Insert the values:
V = 20 * 16 * 12
Multiply the values:
V = 3840in²
Tools -
Question 10
Steven fosses a fair coin 100 times. The coin lands on heads 57 times and tails 43 times.
What is the probability and relative frequency of the coin landing on heads?
A
The probability is 50% and the relative frequency 43%
B
The probability is 50% and the relative frequency 57%
The probability is 43% and the relative frequency 50%.
D
The probability is 57% and the relative frequency 50%.
Answers
Answer: D
Step-by-step explanation:
The probability of the coin landing on heads is 50% and the relative frequency of the coin landing on heads is 57%
The probability of the coin landing on headsThe sample space of a coin is:
S = {Head, Tail}
The sample size is:
n(S) = 2
The probability of the coin landing on heads is calculated as:
P(Head) = n(Head)/n(S)
So, we have:
P(Head) = 1/2
Express as percentage
P(Head) = 50%
Hence, the probability of the coin landing on heads is 50%
The relative frequency of the coin landing on headsFrom the question, we have:
Toss = 100
Head = 57
The relative frequency of the coin landing on heads is calculated as:
Frequency = 57/100
Express as percentage
Frequency = 57%
Hence, the relative frequency of the coin landing on heads is 57%
Read more about probability at:
https://brainly.com/question/251701
Find the perimeter and area of the polygon shown below.
P= 72 feet, A = 300 square feet
O P = 56 feet, A = 308 square feet
P= 87 feet, A = 360 square feet
P = 72 feet, A = 308 square feet
Answers
Answer:
A. P = 72 feet, A = 300 square feet
Step-by-step explanation:
✔️Perimeter = sum of all sides of the polygon = 15 + 16 + 17 + 8 + 16 = 72 ft
✔️Area of the polygon = area of a trapezoid = ½(a + b)h
Where,
a = 16 ft
b = 16 + 8 = 24 ft
h = 15 ft
Substitute
Area = ½(16 + 24)15
Area = ½(40)15
Area = 300 ft²
1. The perimeter of a square box is 12x+32. Write an equivalent expression that shows the perimeter as four times the side length of the box.
Answers
Answer:
4(3x+8)
Step-by-step explanation:
You volunteer to help drive children at a charity event to the zoo, but you can fit only 7 of the 19 children present in your van. How many different groups of 7 children
can you drive?
How many different groups of 7 children can you drive?
Answers
Answer:
3
Step-by-step explanation:
Find the surface area of the prism below.
Square meters
Answers
Answer:
loluuuhsjsjzjjnsjs
Step-by-step explanation:
bzisjsjxjshsusud773938
Brainliest for correct answer, asap please.
Answers
Answer:
Not a function
Not a function
Function
Step-by-step explanation:
The 1st one isn't as clear in the picture, but it looks like the ends of each line overlap the each other.
Find the height of the cone plsss
Answers
Answer: the last one, 31.6
Answer:
the answer is about 31.6
Step-by-step explanation:
the formula for the height of a cone is 3(V/πr^2), so if you plug in the number, you'll get about 31.6.