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.

For example: [math]\displaystyle{ f(x,y) = x^2y^2 + \frac{x}{y} - x }[/math]. To differentiate this function in respect to [math]\displaystyle{ x }[/math] is the same as treating the function as [math]\displaystyle{ f(x) = ax^2 + \frac{x}{a} - x }[/math]. We treat [math]\displaystyle{ y }[/math] and every occurrence of it as some fixed number and proceed with calculations in the same way as differentiating a single variable function. It's not necessary to write down the substitution. In fact, better not do it because if we write [math]\displaystyle{ y^2 = a }[/math] we may also do this [math]\displaystyle{ y = \sqrt{a} }[/math] which ends up being a backwards reasoning. To treat a variable as another variable. We repeat to differentiate in respect to [math]\displaystyle{ y }[/math]. Now the third term is going to be [math]\displaystyle{ \frac{\partial x}{\partial y} }[/math], which results in zero. It's the same as to differentiate a constant, but in this case we are checking that one variable does not depend on the other.

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.