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I have $$ f(x,y) = \cases{ \sqrt{xy}& if $x>0,y>0$ \\ -\sqrt{xy}& if $x<0,y<0$ \\ 0 } $$ I want calculate directional derivative $D_vf(0,0)$ with $v=(1,1)$

f is not differentiable in the origin but do directional derivatives exist ?

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Hint: Your question is: does the limit $\displaystyle\lim_{t\to0}\frac{f(t,t)}t$ exist? What do you think?

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  • $\begingroup$ $\lim_{t\to0^+}\frac{f(t,t)}t=\lim_{t\to0^+}\frac{|t|}t=1$ and $\lim_{t\to0^-}\frac{f(t,t)}t=\lim_{t\to0^-}\frac{-|t|}t=1$ so exist? $\endgroup$ – Giulia B. Nov 26 '18 at 18:33
  • $\begingroup$ Yes, it does, What's the doubt? $\endgroup$ – José Carlos Santos Nov 26 '18 at 18:36

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