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. We have to study how the function behaves for each direction separately from the others. With the axes being linearly independent 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, albeit we are required to rely on linear algebra to do it properly.
Graphically we have this:
See that partial derivatives, graphically, mean that we are considering derivatives parallel to each axis. While we "walk" parallel to an axis we have variations in one direction but not in the others. That's why multivariable calculus requires vectors, because we have multiple directions with multiple variables. Notice that to differentiate in respect to one variable we keep a constant distance from the axis we are parallel to, the distance itself doesn't matter as long as it is a constant. That's the graphical meaning of treating a variable as a constant.
We can easily extend the same limit that we have to define the derivative for a single variable to many variables:
[math]\displaystyle{ \frac{\partial f}{\partial x} (x, y) = \lim_{x \ \to \ p} \frac{f(x, \ y) - f(p, \ y)}{x - p} }[/math] or [math]\displaystyle{ \lim_{h \ \to \ 0} \frac{f(x + h, \ y) - f(x, \ y)}{h} }[/math]
Notice that one variable is kept fixed, we don't have any increments for it. The quotient also has the distance between two points along a straight line, parallel to the axis. Geometrically, it doesn't make sense to try to have increments in both directions at the same time. Because the denominator is a distance between two points for one variable, not two.
