6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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.

Find the slope of the curve y=sin^-1 x at (1/2, π/6) without calculating the derivative of sin^-1 x.

Evaluate the derivative of the following functions.

f(x) = sin^-1 2x

Evaluate the derivative of the following functions.

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

Evaluate the derivative of the following functions.

f(y) = tan^-1 (2y² - 4)

Evaluate the derivative of the following functions.

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

Evaluate the derivative of the following functions.

f(t) = ln (sin^-1 t²)

Evaluate the derivative of the following functions.

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

Evaluate the derivative of the following functions.

f(x) = sec^-1 (ln x)

Evaluate the derivative of the following functions.

f(x) = sin(tan^-1 (ln x))

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

b. d/dx(tan^−1 x) =sec² x

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]

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)=(x²−1)sin^−1 x on [−1,1]

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)=(sec^−1 x)/x on [1,∞)

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) = 1/2x+8; (10,4)