Suppose we go beyond 
and talk about
---
e.g., a function of the exact position in space.
This is just an example, of course; the abstract idea of
a function of several variables can have ``several'' be
as many as you like and ``variables'' be anything you choose.
Another place where we encounter lots of functions of
``several'' variables is in Thermodynamics, but
for the time being we will focus our attention on the three
spatial variables x (left-right), y (back-forth)
and z (up-down).
How can we tackle derivatives of this function?
Well, we do the obvious: we say, ``Hold all the other
variables fixed except [for instance] x and then treat
as a function only of x, with
y and z as fixed
parameters.'' Then we know just how to define the derivative
with respect to x. The short name for this derivative is the
partial derivative with respect to x, written symbolically
where the fact that there are other variables being held fixed
is implied by the use of the symbol 
instead of just d.
Similarly for 
and
.