9–61. Evaluate and simplify y'.


y = 2x² cos^−1 x+ sin^−1 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 = tan^−1 √t²−1

9–61. Evaluate and simplify y'.

y = x²+2x tan^−1(cot 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

Find the derivative of the inverse of the following functions. Express the result with x as the independent variable.

f(x) = x^-1/3

Find the derivative of the given function.

$f(x)=(x^3+4x)\sin^{-1}x$