Go back and look at Newton's and Leibniz's notation. Math notation, proof structure, and simplicity are continually getting cleaned up. The thing is, math is HARD.
I read that Leibniz had studied Chinese and that it influenced some of the math notation that he introduced. Like using df/dx to express the derivative of f with respect to x as a kind of ratio. Whereas Newton just put f' to denote the same thing. I think Leibniz also came up with the elongated sigma for representing integrals as a type of summation.
And Newton's dot notation could only represent differentiation with respect to time. Obviously what he was focused on, sure, but the more general notation was obviously also needed.
Frankly, what annoys me is how mathematicians grab natural language words to represent their concepts, and often pick the worst possible of such words to use. For example: "Imaginary" numbers are no more imaginary than the "real" numbers are. They're just as well-defined, just as practically useful, and at least as interesting. But that idiotic name hangs on them, a relic from a more ignorant era, and scares away who knows how many students.
The term imaginary was not chosen to be descriptive:
> At the time, such numbers were poorly understood and regarded by some as fictitious or useless, much as zero and the negative numbers once were. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory. -- http://en.wikipedia.org/wiki/Imaginary_number
df/dx is actually completely obvious. It tells you exactly what's going on, since df is the change of f and dx is the change of x, for very small changes. It's a lot more precise than simply writing f'. Newton wrote a dot over f and a lot of physicists still do that.
