6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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

23. y=arcsin(√2t)

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

b. (f^-1)'(6)

Angle to a particle (part 2) The figure in Exercise 81 shows the particle traveling away from the sensor, which may have influenced your solution (we expect you used the inverse sine function). Suppose instead that the particle approaches the sensor (see figure). How would this change the solution? Explain the differences in the two answers. <IMAGE>

Derivatives of inverse functions from a table Use the following tables to determine the indicated derivatives or state that the derivative cannot be determined. <IMAGE>

c. (f^-1)'(1)

{Use of Tech} Angle of elevation A small plane, moving at 70 m/s, flies horizontally on a line 400 meters directly above an observer. Let θ be the angle of elevation of the plane (see figure). <IMAGE>

a. What is the rate of change of the angle of elevation dθ/dx when the plane is x=500 m past the observer?

Which of the following is the correct derivative of $y=arcsin\left(x\right)$ with respect to $x$?

Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

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

27. y = (1 - θ)tanh⁻¹(θ)

Evaluate the derivative of the following functions.

f(x) = sin^-1 (e^-2x)

47–56. Derivatives of inverse functions at a point Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

f(x)=tan x; (1,π/4)

13–40. Evaluate the derivative of the following functions.

f(x) = 1/tan^−1(x²+4)

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

37. y=s√(1-s²) + arccos(s)

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot x)

If f is a one-to-one function with f(3)=8 and f′(3)=7, find the equation of the line tangent to y=f^−1(x) at x=8.

9–61. Evaluate and simplify y'.

y = tan^−1 √t²−1