# Directional derivative of a function

Could anyone help me with this?

Which directional derivative exists at the Point $(0,0)$ of function $$f(x,y)=\sqrt{|xy|}$$

Well i made this: Assume $(v,w)$ a vector with lenght $v^2+w^2=1$ and with formula $$\lim_{t\to0} \frac{f(t(v,w)) - f(0,0)}{t} = \lim_{t\to0} \frac{f(tv,tw)-f(0,0)}{t} \\ = \lim_{t\to0} \frac{\sqrt{|tvtw|}}{t} = \lim_{t\to0} \frac{t\sqrt{|vw|}}{t} = \sqrt{|vw|}$$ so Limit of this function at the Point $(0,0)$ exists it means all directional derivatives exist

I'm not sure with this though. Can anyone please revise it if i ever made a mistake!

thanks

• Note that $\sqrt{|tvtw|} = |t|\sqrt{|vw|}$. – Arthur Apr 18 '16 at 17:55
• yeah it is sorry – suugii02 Apr 18 '16 at 17:56