Derive or differentiate?

Solution 1:

In English, I've almost always heard mathematicians say "We now differentiate $f$ to get ...". Occasionally I've heard "derive," but in English (my native language!), that's generally used to mean "work out", as in "Ralph couldn't derive a proof of the intermediate value theorem from the information he had at hand." It's also used in generating one thing from others, as in "We can now derive the half-angle formulas from the addition formula by a clever substitution."

Solution 2:

That is a question for a native speaker, I fear.

In German both are used

• to differentiate = differenzieren (determing the derivative)
• to derive = ableiten -> Ableitung (derivative)

In English literature, I think I only saw differentiate for the operation.

In German you can use "Herleitung" to stress more that it is about taking conclusions. In English it is maybe "derivation".

Solution 3:

In mathematical/Calculus sense only: to differentiate is the verb "to find or calculate the derivative" The noun is "the derivative" Non-calculus students assume that it's ok to say or ask "how do I derive this function?" NO! Calculus students ask, "How do I differentiate this function?" Or "How may I find the derivative of this function?"

Conclusion: right now, in the 21st century Calculus instructors use the verb = differentiate and the noun = derivative