# Derivative of multivariate integral

1. Jul 29, 2014

### supaveggie

1. The problem statement, all variables and given/known data

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

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...

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

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

The solution I have is showing
$\frac{dW}{dσ}= 2π\int_0^∞y'(H(x,σ))\frac{dH(x,σ)}{dσ}x,dx$ 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. Jul 29, 2014

### LCKurtz

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

3. Jul 29, 2014

### verty

The syntax is wrong, there is a missing ). Also, I see no attempt. And what is H?

4. Jul 29, 2014

### supaveggie

updated the original post. Not seeing why the partial was pushed through the function y

5. Jul 29, 2014

### verty

Hint: let $f(x, σ) := y \circ H$.

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