The use of infinitesimals in calculus! Things like integrating against "dx" or taking a derivative of a function "df/dx" make calculus seem like something it's not. Worse, it really introduces a whole new language whose rules for manipulation are unclear.
Like if we have a function f(x,y), we often write the "total" derivative of f as
df = (df/dx) dx + (df/dy) dy
And the rules for manipulating a differential like this are quite strange (e.g., dx on top cannot cancel the dx on bottom).
There's a language of differential forms that makes this notation a bit more rigorous, but I still think the notation is very misleading and confusing, especially for those who begin to learn calculus.
To be fair to mathematics, in your example the proper notation is df = (del f/del x) dx + (del f/del y) dy, which does not suggest the possibility of cancelation.
