f(x) = tan(x) +3x for x in (-pi/2,pi/2)
This is just a little part of the whole question which is on the Inverse Function Theorem, I need to find the inverse to find out if it can be differentiated but can't seem to find the inverse to begin with! Please help!
Prove that the function f(x) = tan x +3x for x in (-pi/2, pi/2) has an inverse function that is differentiable on R, and find the value of the derivative of the inverse at 0.
I have checked that the conditions for the IFT hold and I think they do, so I thought I needed to find the inverse at this point?! Thank you for helping!
The function is invertible because it is strictly increasing, (the derivative is , which is always positive.
The function is differentiable everywhere .
Therefore its inverse will also be differentiable everywhere except where .
If
.
This does not have a solution in the real numbers, so , and so exists and will be differentiable.
The derivative of the inverse is given by the Inverse Function Theorem:
.