Why is $(f(x))'$ shortened $f'(x)$?
This makes the chain rule look awkward, as $(f(g(x)))'\neq f'(g(x))$, but rather $f'(g(x))\!\times\! g(x)$, and makes it difficult to remember.
It's also an awkward way as $(f(x))' = ('\circ f)(x)$, and changing the order of the functions makes no sense, as $'$ should come first.
Calculus seems to be a wonderful source of bad notation; I've never seen the notation $(f(x))'$ used that much. $f'(x)$ captures things better, since the $'$ mark applies to the function, not to the expression as a whole, and is what I usually see - so I would say the $(f(x))'$ is more of a corruption of $f'(x)$ than the other way round. I'd generally state the chain rule as: $$(f\circ g)'(x) = f'(g(x))g'(x)$$ which looks a lot clearer.