Is that directional derivative OK?

Question

Recently I solved a directional derivative and I think it’s okay but I’m not sure. The question of the exercise is: find the directional derivative at P=(1,1) of f(x,y)=|x+y-2| I substituted and did the math, and it gave me cos(θ)+sin(θ). Thanks in advance.

Answer 1

The directional derivative along $v=(\cos \theta, \sin \theta)$ would be given by \begin{align*} f'_v(1,1)= &\lim_{t \to 0}\dfrac{f(1+t \cos \theta, 1+ t \sin \theta)-f(1,1)}{t}=\lim_{t\to 0} \dfrac{|1+t\cos \theta+1+t\sin \theta-2|}{t}\\ =& |\cos \theta+\sin \theta| \lim_{t\to 0}\frac{|t|}{t}. \end{align*}

But, since this limit does not exist, the directional derivative is not defined at the point.