Calculating partial derivative for a function defined by an integral

Question

I have no idea how to solve the following problem. Please suggest some suitable solutions.

Define $$f(x,y)=\int_0^{\sqrt{xy}} e^{-t^2} \,dt,$$ for $x>0, y>0$. Compute $\dfrac{\partial f}{\partial x}$ in terms of $x$ and $y$.

Answer 1

Note that for each fixed $y$, you can write $f$ in the form $f(x,y)=\int_0^{g(x)}e^{-t^2}dt$ for some function $g$ that only depends on $x$. Then, you can apply the fundamental theorem of calculus and the chain rule to get what you want.

Answer 2

Use the chain rule: $$\frac{\partial f}{\partial x}\int_0^{\sqrt{xy}} e^{-t^2} \,dt=\left(\frac{\partial f}{\partial x}\sqrt{xy}\right)\cdot e^{-(\sqrt{xy})^2}=\frac{y}{2\sqrt{xy}}e^{-xy}$$