> "what's so special about x ?" Does f(x) mean one function, whereas f(y) means a completely different function?

This looks like the difference between parameters of a function and arguments. In the definition, you have the parameter x, used internaly and, when calling the function you use an argument - located in the calling context. In python: def f(x): return 2*x ## x is a parameter x = 3 y = f(x) # x in an argument

For partial derivatives, it may be ok to use 1 and 2 to show they are the first and second parameter, but maybe they could also be named by some convention, like x and y or alpha and beta or whatever