Fundamental theorem of calculus with 2-variables integrand

Question

I have an equation in the form

$$\int_x^\infty f(x,y) \, dy = \int_x^\infty g(y) \, dy$$

from which I would like to derive a relation between the functions $f$ and $g$.

If I take the partial derivative w.r. to $x$ of both sides, I get, using the fundamental theorem of calculus,

$$\partial_x \left( \int_x^\infty f(x,y) \, dy\right) = -g(x)$$

However, I don't know what to do with the first term.

Is there anything I can conclude about it without having to find an analytical expression for the integral?

Answer 1

You can employ chain rule in two dimensions. Define $$h(x,z)=\int_z^\infty f(z,y)\, dy$$ and $$w(x)=(x,x).$$ Then you have $$\int_x^\infty f(x,y)\, dy = h(w(x)),$$ hence you can derive the composition on the right hand side by chain rule: $$\partial_x\left(\int_x^\infty f(x,y)dy\right)= -f(x,y)+\int_x^\infty\partial_x f(x,y)dy$$