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Thread: FTOcalculus simplification problem

  1. #1
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    FTOcalculus simplification problem

    d/dx integral (f^-1 (t) dt) the boundaries are 0 to f(x).
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  2. #2
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    Quote Originally Posted by myoplex11 View Post
    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,
    $\displaystyle g'=xf'$
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