We have a formula for the derivative of an inverse function:
dy/dx = 1/(dx/dy).
Just how useful is it?
Say we want to find the inverse of a complicated function, f(x), on an interval (a,b) on which f(x) is one-to-one. Can we use integration to find such a function?
Example: Say we didn't know much about the function h(x) = sin(x), but wanted to express its inverse as an integral (this was my inspiration for the idea). How could this be done?
More importantly, this would apply to functions like F(x) = x*e^x. Its inverse, W(x), is important in several applications. Say I choose the branch on (0, infinity). Could I express this branch (or any other I choose) as an integral of well-defined functions?