6. Derivatives of Inverse, Exponential, & Logarithmic Functions

How are the derivatives of sin^−1 x and cos^−1 x related?

Tangent lines Find an equation of the line tangent to the graph of f at the given point.

f(x) = sin⁻¹(x/4); (2,π/6)

Tangent lines Find an equation of the line tangent to the graph of f at the given point.

f(x) = sec⁻¹(e^x); (ln 2,π/3)

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

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>

a. (f^-1)'(4)

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)

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)

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>

d. f'(1)

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)