Differentiation and integration

[davon806]

Homework Statement

Hi,I saw a statement in my physics notes like this(Anyway it is a maths problem):

where L is a general differential operator.G is a green's function(I guess it is irrelevant)
My question is related to the red line:
Suppose we have this:
∂/∂x ∫ f(x-y)g(y) ∂y
is it generally true that
∂/∂x ∫ f(x-y)g(y) ∂y = ∫ ∂/∂x [ f(x-y)g(y) ] ∂y ?

Homework Equations

Please answer it as simply as you can...Since I have not done multivariable calculus...(Though I would be happy to check it out if there is a theorem related to this.)

The Attempt at a Solution

If the above is simplified to ∂/∂x ∫ f(x)g(y) ∂y ,then ∂/∂x [f(x) ∫ g(y) ∂y]
⇒ ∫ ∂/∂x [ f(x) ] g(y) ∂y
⇒ ∫ ∂/∂x [ f(x)g(y) ] ∂y

But I don't know what to do with the (x-y) term..

Thanks

 

[ShayanJ]

The general formula is:

But you don't need this. You only need to use the properties of Dirac delta, i.e. $\int \delta(x-y) f(y) dy=f(x)$.

 

[davon806]

The general formula is:

But you don't need this. You only need to use the properties of Dirac delta, i.e. $\int \delta(x-y) f(y) dy=f(x)$.

Thanks :) Is there a name for this formula?

 

[ShayanJ]

Leibniz integral rule