6. Derivatives of Inverse, Exponential, & Logarithmic Functions

Which of the following is the correct derivative of $y=arcsin\left(x\right)$ with respect to $x$?

In Exercises 25–36, find the derivative of y with respect to the appropriate variable.

27. y = (1 - θ)tanh⁻¹(θ)

Find the derivative of the given function.

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

Find the derivative of the given function.

$y=4\arccos\left(3x^6-x^5\right)$

Find the derivative of the given function.

$f(x)=\tan^{-1}(x^2)$

Find the derivative of the given function.

$f(t)=\csc^{-1}(2t+7)$

Find the derivative of the given function.

$f\left(x\right)=\cot^{-1}\left(e^{\cos x}\right)$

Find the slope of the line tangent to the graph of y = sin^−1 x at x=0.

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

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)