Partial derivatives and direction

The idea of partial derivatives is pretty similar to the regular derivative. The concept of a derivative is that of a rate of change. For multivariable functions we have to look for rates of change on a per variable basis. That's the meaning of "partial". A multivariable function can increase in one direction and decrease in another. It's impossible for a function to both increase and decrease in the same direction. With the axes being linearly independent from each other we can differentiate in respect to one variable, while the others are treated as constants. The same discussion that we make about conditions for differentiability for a single variable can be made for many variables.

Graphically we have this: