# FTOcalculus simplification problem

• Dec 4th 2006, 01:47 PM
myoplex11
FTOcalculus simplification problem
d/dx integral (f^-1 (t) dt) the boundaries are 0 to f(x).
• Dec 4th 2006, 04:31 PM
ThePerfectHacker
Quote:

Originally Posted by myoplex11
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,
$\displaystyle g(x)=\int_0^{f(x)} f^{-1}(t)dt$
Then you can view $\displaystyle g$ as a composition,
$\displaystyle h\circ f$
Where,
$\displaystyle h=\int_0^x f^{-1}(t)dt$
Thus,
$\displaystyle h$ is differenciable.
Thus we can use chain rule theorem,
$\displaystyle g'=f'\cdot (h'\circ f)$
But by fundamental theorem,
$\displaystyle h'=f(x)$
Thus,
$\displaystyle g'=f'\cdot (f\circ f^{-1})$
But,
$\displaystyle f\circ f^{-1}=x$ the identity map.
Thus,
$\displaystyle g'=xf'$