The origin/use of "derivative" and "differentiate"

Solution 1:

In French there is less of a problem with terminology: differentiation is called "dérivation", and derivative is, you guessed it, "la dérivée". The term "fonction dérivée" was originally introduced by Lagrange. The English term "differentiate" ultimately derives from "differences"; namely, Leibniz originally studied finite differences and discovered certain patterns that led him to introduce (infinitesimal) differentials.

Solution 2:

I believe the term "derivative" arises from the fact that it is another, different function $f'(x)$ which is implied by the first function $f(x)$. Thus we have derived one from the other. The terms differential, etc. have more reference to the actual mathematics going on when we derive one from the other.