Derivative of multivariate integral

[supaveggie]

Homework Statement

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

Homework Equations

both y and H(x,y) are continuous functions from 0 to Infinity

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

 

[LCKurtz]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Tried using the leibniz rule but it's not really getting me anywhere...

Thanks

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

 

[verty]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Tried using the leibniz rule but it's not really getting me anywhere...

Thanks

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

 

[supaveggie]

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

 

[verty]

Hint: let $f(x, σ) := y \circ H$.

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