Fast example: f(x) = x^2.

The derivative is df/dx = 2x, the *expression* for the rate of change of f with respect to x.

Differentiation is the *process* you did to get it, where you multiply by the exponent and then reduce the exponent by one. So you use differentiation to obtain derivatives, just like you use integration to obtain integrals or you use baking to obtain a cake.

“Derivative” is, for lack of a better term, a slang term that has come into common use.

The thing you do where you find the slope of the tangent line to a function is called *differentiation*. The result was originally a *differential* (that term is still in use in mathematics, but has a number of specific meanings that only partially overlap with modern *derivative*). The word comes from the word *difference*, since the process of differentiation ultimately involves measuring very small differences in function values.

(Of course, the actual word first used by Leibnitz was actually *differentialis*, since he wrote in Latin.)

The word “derivative” comes from the word “derive”, and originally referred to any function that was derived somehow from another function; that is, it did not exclusively refer to functions derived from others through the process of infinitesimal differentiation. I’ve read that the first use of the word “derivative” to refer to the result of infinitesimal differentiation was Lagrange’s 1797 treatise on analytic functions. Note that at this point, calculus^1 had been around for more than a century.

(Of course, Lagrange mostly wrote in French, so the word he used was actually *la dérivée*, short for *la fonction dérivée*, “derived function”. Note that the French word used for *differentialis* was *la différence*, or just English “difference”).

I haven’t done a thorough study, but from what I can tell *dérivée* started becoming more and more popular in French, maybe because it’s shorter than *différence* or because it clearly doesn’t refer to the result of “ordinary subtraction”, maybe just because French mathematicians low-key worshipped Lagrange. Anyway, by the early 19th Century, *dérivée* was standard in French and referred unambiguously to the modern sense (unless otherwise specified). From French, “derivative” was adopted more and more into English as well, the French being the foremost mathematicians in the world at the time.

I can say for certain that by the late 19th Century, “derivative” was used in English, but I don’t know if it was being used exclusively (or even primarily), or for how long it had been used. In the well-known 1910 textbook Calculus Made Easy by Silvanus Thompson, he writes

>The correspond symbol for the differential coefficient is f'(x), which is simpler to write than dy/dx. This is called the “derived function” of x.

To me, that suggests that “derivative” (or variants) were in use, but probably not exclusively so. But that’s just my interpretation.

1. Note that before the Calculus of Differential and Summation, “calculus” was a generic term referred to a general system of methods in mathematics)

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