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Derivative of multivariate integral

  1. Jul 29, 2014 #1
    1. The problem statement, all variables and given/known data

    Trying to figure our how to solve the following: [itex]\frac{dW}{dσ}[/itex]
    where [itex] W(σ) = 2π\int_0^∞y(H(x,σ))x,dx [/itex]

    2. Relevant equations
    both y and H(x,y) are continuous functions from 0 to Infinity

    3. The attempt at a solution
    Tried using the leibniz rule but it's not really getting me anywhere...

    [itex]\frac{dW}{dσ} = 0+0+2π\int_0^∞\frac{\partial(y(H(x,σ))x)}{\partial \sigma},dx[/itex]

    I'm not familiar with a chain rule for partial differentiation...

    The solution I have is showing
    [itex]\frac{dW}{dσ}= 2π\int_0^∞y'(H(x,σ))\frac{dH(x,σ)}{dσ}x,dx [/itex] I'm not understanding how they arrived at this.
    It is also unclear what y' represents as ' is not necessarily used for derivative or defined anywhere...
    Thanks
     
    Last edited: Jul 29, 2014
  2. jcsd
  3. Jul 29, 2014 #2

    LCKurtz

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    Assuming nice enough conditions on H and the convergence of the improper integral (which I haven't checked), I would expect$$
    W'(\sigma) = 2π\int_0^∞y\frac{\partial(H(x,σ)}{\partial \sigma}x,dx$$
     
  4. Jul 29, 2014 #3

    verty

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    The syntax is wrong, there is a missing ). Also, I see no attempt. And what is H?
     
  5. Jul 29, 2014 #4
    updated the original post. Not seeing why the partial was pushed through the function y
     
  6. Jul 29, 2014 #5

    verty

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    Hint: let ##f(x, σ) := y \circ H##.

    I think this is about as much help as I can give. Best of luck.
     
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