Switching Orders of Differentiation and Integration in case of Function of 2 Variables

Question

I encountered the following equation in my calculus lecture notes: $$ \frac{d}{dx}\int_0^xf(x,y)dy = f(x,x)+\int_0^x\frac{\partial f}{\partial x}(x,y)dy $$ Could you show me how to prove the above equality?

Answer 1

This is a particular version of what is known as the Leibniz Integral Rule. The wikipedia link provides a nice proof of the general case(s), and it is not too hard to see how this particular form follows