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Math Help - Prove the following function is differentiable

  1. #1
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    Prove the following function is differentiable

    Let f(t) be a continuous function under 1 variable, and we shall define

    g(u,v)=Integral of f(t)dt from (v^2 - u^2) to (v^2+u^2)

    prove that g(u,v) is differentiable in (u,v).


    How do I approach? Should I use the Leibniz rule for integrals?

    Thanks.
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  2. #2
    Senior Member roninpro's Avatar
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    Is your integral this?

    \displaystyle g(u,v)=\int_{v^2-u^2}^{v^2+u^2} f(t)\text{ d}t

    Did you try using the Fundamental Theorem to compute g_u and g_v?

    Fundamental theorem of calculus - Wikipedia, the free encyclopedia
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  3. #3
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    Which is, essentially, the Leibniz formula GIPC refers to.
    \frac{\partial g}{\partial u}= f(v^2+ u^2)(2u)- f(v^2- u^2)(-2u)
    \frac{\partial g}{\partial v}= f(v^2+ u^2)(2v)- f(v^2- u^2)(2v)
    so not only do the partial derivatives exist but, since we are given that f is continuous, they are continuous functions and so g is differentiable.
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