8. Derivatives of Inverse and Inverse Trig Functions

Inverse functions Inverse functions are functions that have x and y points switched Therefore, at corresponding points, they have reciprocal slopes

Derivative of an inverse function If f(x) is a function and g(x) is its inverse, and f(a)=b and g(b)=a then b is a y value of the original function and a is the corresponding x value

Example 1 If , find f(1), f’(1), f-1(3), and (f-1)’(3)

Example 2 If , find (f-1)’(2)

Example 3 If m(x) and n(x) are inverses and if m(5) = -9 and m’(5)=-8/11, find n’(-9)

Derivatives of inverse trig functions

Example 4 Find the equation of the tangent line to the graph of at x = 1.

Example 5 Calculate f’(1/2) where f(x) = arcsin(x2)

Example 6

Example 7 Differentiate then simplify to a single term

Example 8 Find the coordinate of any horizontal tangent and find the equation of any horizontal asymptotes of the derivative of