My task was to find the directional derivative of function:
$$z = y^2 - \sin(xy)$$ at the point $(0, -1)$ in direction of vector $u = (-1, 10) $.
The result I found was $-21/\sqrt{101}$. But I can't figure out what is the interpretation of this result.
Does it mean that the function grows fastest with that derivative or with something else?
On
The directional derivative is a Linear Operator that best approximates (at the first order) the growth of a function on a given direction:
$$\frac{\vert\vert f(x+h)-Lh\vert\vert}{h} \rightarrow 0 \, \text{as} \, h\rightarrow 0$$
Where $h=(h_1,h_2) $ is the increment on your direction, that in your case will be $h=ku$ with $k \rightarrow 0$
You are standing on a hill side. If you travel in one direction you go up the hill, another direction you go down the hill and yet another direction you go across the hill.
The directional derivative tells you the steepness of the slope if you walk in the indicated direction.