How can you find the tangent plane to a given point on a surface? (Verbal descriptions preferred)
I'm thinking you can find the "vector versions" of two directional derivatives (maybe the partial derivatives $f_x$ and $f_y$ would be simplest) and then take their cross product, and that gives you a normal vector to the tangent plane. But (1) how would you write said partial derivatives as vectors, and (2) are there other, more convenient ways of finding the plane?
(Saw some very similar questions posted, but they were all about specific problems and I didn't really get the explanations)