6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Find the slope of the line tangent to the graph of y = tan^−1 x at x= −2.

For problems 49–52 use implicit differentiation to find dy/dx at the given point P.

51. y arccos(xy) = -3√2/4 π; P(1/2, -√2)

Suppose the slope of the curve y=f^−1(x) at (4, 7) is 4/5. Find f′(7).

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

21. y=arccos(x²)

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

62–65. {Use of Tech} Graphing f and f'

b. Compute and graph f'.

f(x)=e^−x tan^−1 x on [0,∞)

128. Derive the formula dy/dx = 1/(1+x²) for the derivative of y = arctan(x) by differentiating both sides of the equivalent equation tan(y)=x.

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

39. y=arctan√(x²-1) + arccsc(x), x>1

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

41. y= x arcsin(x) + √(1-x²)

Evaluate the derivative of the following functions.

f(y) = tan^-1 (2y² - 4)

62–65. {Use of Tech} Graphing f and f'

c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.

f(x) = (x−1) sin^−1 x on [−1,1]

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) = 3x-4

Find (f^−1)′(3), where f(x)=x³+x+1.

Suppose f is a one-to-one function with f(2)=8 and f′(2)=4. What is the value of (f^−1)′(8)?

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

f(x) = x^-1/3