6. Derivatives of Inverse, Exponential, & Logarithmic Functions

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

35. y=arccsc(e^t)

10–19. Derivatives Find the derivatives of the following functions.

f(t) = cosh t sinh t

Evaluate the derivative of the following functions.

f(u) = csc^-1 (2u + 1)

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

33. y=ln(arctan(x))

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

29. y=arcsec(1/t), 0<t<1

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

27. y=arccsc(x²+1)

Find the derivative of the given function.

$f(t)=\(\csc\)^{-1}(2t+7)$

Tracking a dive A biologist standing at the bottom of an 80-foot vertical cliff watches a peregrine falcon dive from the top of the cliff at a 45° angle from the horizontal (see figure). <IMAGE>

b. What is the rate of change of θ with respect to the bird’s height when it is 60 ft above the ground?

67–78. Derivatives of inverse functions Consider the following functions (on the given interval, if specified). Find the derivative of the inverse function.

f(x) = x^2/3, for x>0

10–19. Derivatives Find the derivatives of the following functions.

f(x) = tanh⁻¹(cos x)

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

17. y = ln(arccos(x))

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

35. y = sinh⁻¹(tan x)

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

21. y = z arcsec(z) - √(z² - 1), z>1

Evaluate the derivative of the following functions.

f(x) = x² + 2x³ cot^-1 x - ln (1 + x²)

10–19. Derivatives Find the derivatives of the following functions.

f(x) = (sinh x) / (1 + sinh x)