General Mechanics/Partial Derivatives
Partial Derivatives 
In one dimension, the slope of a function, f(x), is described by a single number, df/dx.
In higher dimensions, the slope depends on the direction. For example, if f=x+2y, moving one unit x-ward increases f by 1 so the slope in the x direction is 1, but moving one unit y-ward increases f by 2 so the slope in the y direction is 2.
It turns out that we can describe the slope in n dimensions with just n numbers, the partial derivatives of f.
To calculate them, we differentiate with respect to one coordinate, while holding all the others constant. They are written using a ∂ rather than d. E.g.
Notice this is almost the same as the definition of the ordinary derivative.
If we move a small distance in each direction, we can combine three equations like 1 to get
The change in f after a small displacement is the dot product of the displacement and a special vector
This vector is called the gradient of f. It points up the direction of steepest slope. We will be using this vector quite frequently.
Partial Derivatives #2 
Another way to approach differentiation of multiple-variable functions can be found in Feynman Lectures on Physics vol. 2. It's like this:
The differentiation operator is defined like this: in the limit of . Adding & subtracting some terms, we get
and this can also be written as
Alternate Notations 
For simplicity, we will often use various standard abbreviations, so we can write most of the formulae on one line. This can make it easier to see the important details.
We can abbreviate partial differentials with a subscript, e.g.,
Mostly, to make the formulae even more compact, we will put the subscript on the function itself.