If y=cosh(4tanh¯¹x) then dy/dx=​

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]

A cord of length 71.5 m has been cut into 26 pieces of equal length. What is the length
of each piece?

The volume of a rectangular prism is
represented by the expression x3 + 5x2 - 8x -
48 If the length is (x – 3) and the height and
width are equal, what is the width of the prism?
(x + 4)
(x-4)
(x - 2)
(x + 2)
plsss hurry

hurry fast ill give brainlyist\
9+10

A fair coin has a probability 0.5 of coming up heads. If you toss a fair coin twice, are you certain to get one head and one tail?

A volunteer at the zoo is responsible for feeding the animals in 15 exhibits in the reptile house. This represents 20% of the total exhibits in the reptile house. How many exhibits are in the reptile house.

Write an equation for the description.

The length of a rectangle is twice its width. The perimeter of the rectangle is 130 feet.

I need help anyway can help me

Somebody answer how does financial services eg banking contribute to the economy?

Solve using elimination.

–8x + 6y = –4
2x + 2y = –20

between −20/3 and 51/4?

HELPPP PLEASE!!! IM BEING TIMED!!!

Please help late assignments

HELP PLZ !

Plot the numbers 3.07 , 2.92 and 3.02 on the number line

Find the median of the following set of numbers.
{36, 16, 24, 33, 49}

find the answer to 6729 x 92740 x 268​

Volume Round Robin Review
*12
The volume of a cylinder is 301.44 cubic inches. If the area of the base is 37.68 square
inches, what is the height of the cylinder?

what is the probability of rolling a product less than 5 on a pair of 6-sided dice

Solve the following systems of equations. (a) x+y = 5 x-y = -1

Which expression is equivalent to 1/4(5x+6)?
Answer choice :A: (5 1/4x) + (6 1/4x) B: 3/4x + 6/4 - 1/4x C:3/4x + 3/4 + 3/4 + 2/4x D: (1/4x +5) + 1/4x + 6

PLEASE HELP NO LONKS AND PLEASE HURRY

Consider the inequality shown
4x+ax-14>-24
What is the highest integer value of a when x=-2

Town B is 8 miles north and 17 miles west of town A. How far are
the two towns apart?

On November 1, Jovel Company loaned another company $100,000 at a 6% interest rate. The note receivable plus interest will not be collected until March 1 of the following year. The company's annual accounting period ends on December 31, and adjustments are only made at year-end. The adjusting entry needed on December 31 is:

Please show step by step. I will chose Brainlyest once it lets me. 5 s and thanks too. Please answer ASAP.

Simplify each radical 1-4

[
(
x
2
+
5
)
−
14
]
⋅
3
=
45

HELP ASAP!!!!!!!!!!!!!!

I NEED HELP WITH THESE MATH PROBLEMS. I put 30 points for this.
1. The go-cart traveled at 10 miles per hour for half an hour. How far did it travel? show your work.
2. Gooey Good Bakery can bake 16 cakes in 1 hour. How many cakes can the bakery make in 5 3/4 hours? show your work.
3. Jennifer is adding a border to the quilt she made. Her quilt is 6 feet long and 5 feet wide. With an extra 6 inches to overlap, how much border material will she need? show your work.
4. The Marvelous Marble Company makes 100 marbles in 2 hours. How many hours would it take the company to make 1000 marbles? show your work.
5. A feather floats to the ground at a speed of 2 centimeters per second. Find the distance the same feather travels if it floats down for 20 seconds. show your work.

Hey ! Can anyone help me with this problem please thank you !!

Will mark BRAINLIEST IF U ANSWER!: