Thread: Showing that the directional derivative exists at a point.

Show that for any unit vector , the directional derivative at in the direction of exists for the function .

Okay, so maybe I'm just rusty on my multivariable calculus, but aren't each of the partial derivatives undefined at ? What am I missing here? And I'm missing how I would prove whether or not the function is actually differentiable at .

Edit: So I realized I had to use the limit definition of directional derivative, and I got for a unit vector the directional derivative at is and since the gradient is undefined at , the gradient dotted with the unit vector is not so the function is not differentiable at . Is this right? And for a second part where instead, I get that the directional derivative is and that once again, the gradient is undefined at , so the gradient dotted with the unit vector is not , so the function is not differentiable at .

Remember the definition of directional derivative in the direction where in your case is and now just substitute and evaluate.

Another thing is that this particular function gives a counterexample to the converse of the theorem in your other thread.