9–61. Evaluate and simplify y'.


y = tan^−1 √t²−1

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>

Tangents and inverses Suppose L(x)=ax+b (with a≠0) is the equation of the line tangent to the graph of a one-to-one function f at (x0,y0). Also, suppose M(x)=cx+d is the equation of the line tangent to the graph of f^−1 at (y0,x0).

c. Prove that L^−1(x)=M(x).

{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?

{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>

b. Graph dθ/dx as a function of x and determine the point at which θ changes most rapidly.

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot x)

9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

Evaluate d/dx(x sec^−1 x) |x = 2 /√3.

Consider the following functions. In each case, without finding the inverse, evaluate the derivative of the inverse at the given point.

y =√x³+x−1 at y=3