6. Derivatives of Inverse, Exponential, & Logarithmic Functions

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

31. y=arccot(√t)

Newton’s method Use Newton’s method to find all local extreme values of ƒ(x) = x sech x.

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)

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

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

47. y=(arccot(x³))³

9–61. Evaluate and simplify y'.

y = 2x² cos^−1 x+ sin^−1 x

Evaluate the derivative of the following functions.

f(t) = ln (sin^-1 t²)

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)=4e^10x; (4,0)

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

25. y = sinh⁻¹(√x)

Critical points Find the critical points of the function ƒ(x) = sinh² x cosh x.

Find the derivative of the given function.

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

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) = e^3x+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>

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

Find the derivative of the given function.

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