And why is the verb for "take the derivative" "differentiate"? Both words make sense individually to describe things -- df/dx is derived from f by taking differences -- but it doesn't make sense that we use both words.

I had an abstract algebra professor, in my first year of grad school in math, who referred to operators D that satisfy the product rule D(fg) = D(f) g + f D(g) , which are not necessarily defined in terms of a limit, as "derivations". This is a natural generalization of the derivative, and apparently "derivation" is the correct technical term, but for a week or so I thought he was just not speaking English properly.

I'm not an algebraist now.