I calculated the partial derivatives for $f(x,y)=x*y$ and I got that for x it's $y$ and for y it's $x$, pretty simple. Then, at point $(2; -3)$ I get that the partial derivative with respect to x is -3 and with respect to y it's 2. How can I interpret this?
Can I say that for each unit x increases, z decreases 3? And that for each unit y increases, z increases 2? Is that the right interpretation?
Think of $z=f(x,y)$ as a topographical map.
Pretend you're standing at the point $(2,-3)$ (the black dot). If you start walking East, you'll be going down a slope that is approximately $3$ feet down for every $1$ foot you walk Eastward. If instead you started walking North, you'd be going up approximately $2$ feet up for every $1$ foot you walk North.
But then you might wonder what the slope is if you move in some other direction. For that you'd use the directional derivative. I'll spare you the derivation but it turns out the slope as you head in the direction of $\vec v = a\hat e_i + b\hat e_2$$^\dagger$ is approximately $\frac{-3a+2b}{\sqrt{a^2+b^2}}$.
$\dagger$: $\hat e_1$ means walking $1$ foot East and likewise $\hat e_2$ is walking $1$ foot North. So if, for instance you wanted to go $60^\circ$ East of South, using a little trigonometry you'd find that that would be going in the direction of $\vec v = \sqrt{3}\hat e_1 + \hat e_2$.