d/dx integral (f^-1 (t) dt) the boundaries are 0 to f(x).
What makes so angry these problem do not mention intervals of a bijection map, continuity, definess, differenciability,.... Makes me so angry. I JUST CANNOT DO PROBLEMS LIKE THIS. So I am going to assume you have a differenciable and invertible function over the entire number line.
I presume you mean,
Then you can view as a composition,
Where,
Thus,
is differenciable.
Thus we can use chain rule theorem,
But by fundamental theorem,
Thus,
But,
the identity map.
Thus,