# How to examine if multivariable functions are differentiable?

How to examine if functions:
$f(x,y)=|x+y|$
and
$g(x,y)=\sqrt{|xy|}$
are diffirentiable in points: $(0,0)$ for $f(x,y)$ and $(0,1)$ for $g(x,y)$

For example, in the first of your two cases, you may draw on your knowledge of what happens in the one-variable case and remember that $x \mapsto \lvert x \rvert$ is not differentiable in the point $0$ to come up with the guess that that something similar is probably also the case in two variables. More precisely, you can prove that $f$ is not differentiable in $(0,0)$ by approaching $0$ along the $x$-axis, letting $y= 0$ and obtaining two different results for the limits of the difference quotients (so that the limit does not exist).