Find the derivatives of each of the functions in Exercises 51-62. In some cases it may be convenient to do some prelim- inary algebra. (These exercises involve hyperbolic functions and their inverses.) 51. f(x) = x sinh x3 52. f(x) = x sinh° x 53. f(x) = cosh(In(x² + 1)) 54. f(x) = 3 tanh?e* tanh ĩ 55. f(x) = /cosh² x+1 56. f(x) = Vsinh x 58. f(x) = tanh (tanx²) 57. f(x) = sinh¯'(x³) sinh-1x 59. f(x) = 60. f(x) = x/tanh¬1x cosh- 61. f(x) = sin(esi nh" x) 62. f(x) = cosh-'(cosh¬1 x)

Q: Find all solutions to tan(x)=sec(x) that lie in the domain [0, 2)

A: tan(x)=sec(x) Given interval is [0, 2).

Q: 1. Find the derivative 4 ln 2) ³ 3x³ sin(2x) 5x2+3

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Q: 4. Determine the 75th derivative of each of the following functions. f(x) = 3 cos 2x b)

A: given function is f(x)=3cos(2x) we know d dx(cos(ax))=-asin(ax)  and d dx(sin(ax))=acos(ax)

Q: True or false: For any constant C, d [tan(x) + C] = [sec(z)]²

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Q: Find the derivative of each function. a) f(x) = Vx2 +1

A: take derivative

Q: How do you know that the functions sin x, ln x, and ex are not polynomials? Give reasons for your…

A: Polynomials have finite terms. And it has a highest degree.  But sinx , lnx and ex have infinite…

Q: 1. a) Copy and complete the table below: Original Function First derivative Second Derivative f(t)=…

A: Original Functionf(t)=5t−3−5e0=5t3−5f(z)=ln100h(p)=−2e1−pFirst Derivative…

Q: 1) Find the derivative of y = e" tan x

A: As we know that; The chain rule; fgx'=f'gx·g'x And ddxex=exddxtanx=sec2x Given: y=etanx

Q: 33-34 For what values of x does the graph of f have a horizontal tangent? Hercnt 33. f(x) = x + 2…

A: we have f(x) = x+2sin x

Q: If f(x)- 3 cos 4x- - {).₁ 0 -3 -12 4 find f'

A:

Q: 1. If y = √√√x. Find y" X₁

A: To find the higher derivative of y(x).

Q: e² tan (e²) b) dr I sec2 (e-) – 7

A:

Q: cos(2x)–1 lim x'+ 5x ? 3

A:

Q: What is the derivative of sin(2x) at the point (0,0)? O 2 O -2 O 4

A:

Q: Find the derivative of each function. Show all your work and simplify the result completely.…

A: Since you have posted multiple questions, we will solve the first question for you. If you want any…

Q: Use the manual method of your choice (using methods only up to and including section 3.10) to find…

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Q: show that there is a root of sin(x2)=e2-2.

A: To show that there is a root of sin(x2)=e2-2.

Q: 1. Find each derivative. a) y = sin- X c) f(x) = ln- sin x COS X b) y = cos(x² - 2x-8)

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Q: You can use the Law of Cosines (see Figure 3.41) to find an equation that relates and 0. Figure 3.41…

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Q: Exercise 2: Derive the formula for the derivative of arcsec(x) in two different ways: (a) Let…

A:

Q: Consider the functions h : R→R and g : R -R defined by h(x) = 72 (5 + arctan(sinh 4æ)), . g(x) = 144…

A: Using inverse trigonometric  function properties

Q: Example 2: Plot the function f(x)=cos3x below 0< <2

A: