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I need to find all the directions in which the directional derivative exists for the function $f(x,y)=|2x+y|$ at the point $(0,0)$ and their values.

So I used: $$ D_v(f)=\lim_{t\to 0}\frac {f(0+tu) - f(0)}{t} = \lim_{t\to0} \frac{|t|*|2v+w|}{t} $$

Where $u=(v,w)$ means all the generic directions, but the thing is, for me this limit will only exist if $2v+w=0$, now I teoretically got the directions, but how can I calculate the value of the derivatives. I am not even sure that my result is correct, if it is, how can I find the values of these derivatives. (I know the procedure I should use, but here I am having trouble finding the partial derivatives of this function at the origin, I personally think that all directional derivatives are equal to zero, but I don't know how to prove it.

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Directional derivative of $f$ at $(0,0)$ in the direction $(v,w)$ can be understood as follows:

  1. form the composition $g(t) = f(0+tv,0+tw)$
  2. study the existence of $g'(0)$.

In your case, $g(t) = |(2v+w)t| = |2v+w| |t|$. Is this differentiable at $0$?

  • Yes when $2v+w=0$ (why?)
  • No when $2v+w\ne 0$ (relate to $h(t)=|t|$, a canonical example of a nondifferentiable function)
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