Find $\frac{\partial f}{\partial y}(2,1)$ given $f(x,y)= \sqrt{xy + \frac{x}{y}}$

Question

Is there any shortcut that I can take to find this partial derivative? When I try to find the limit by definition I end up having to apply L'Hôpital's rule, with ghastly results. Can I just differentiate $f$ w.r.t. $y$ taking $x$ as a constant? Would I then have to I justify that $f$ is differentiable in $D_{\delta}(2,1)$?

Answer 1

Can I just differentiate $f$ w.r.t. $y$ taking $x$ as a constant?

Yes, this is by definition of partial derivatives. Although I would first deal with question below to justify partial derivative existence and then compute it.

Would I then have to I justify that $f$ is differentiable in $D_{\delta}(2,1)$?

Yes. But this is pretty easy as $f$ can be written as compositions of differentiable maps. You need to justify that what is under the square root is defined and strictly positive. Which is not an issue around the point $(2,1)$.