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. The equation that describes each point of a two variable function has [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math] and when we differentiate in respect to one variable, the other is assumed to be some fixed number (not the zero!). So, we have the domain of the function has a range of accepted values for each variable: [math]\displaystyle{ x = 1 }[/math], [math]\displaystyle{ x = 2 }[/math], [math]\displaystyle{ x = 3 }[/math], ... and the same for [math]\displaystyle{ y }[/math]. Everywhere in the function where some [math]\displaystyle{ y }[/math] shows up and we are differentiating in respect to [math]\displaystyle{ x }[/math] we treat [math]\displaystyle{ y }[/math] as some constant and calculate derivatives in the same way we'd do for a single variable function.

For example: [math]\displaystyle{ f(x,y) = x^2y^2 + \frac{x}{y} - x }[/math].