# Why is differentiation called differentiation?

What is the etymological link between the word 'differentiation' and the procedure it describes?

• I suspect that its called differentiation as it is the dual of integration. – Q the Platypus Sep 22 '16 at 3:58
• Why would "differentiation" be the dual of "integration" (as a word)? Also, the before needs to be proven and does not necessarily hold for any integration theory (it does for the Riemann integration). – gented Sep 22 '16 at 8:31
• The result of differentiation is called the derivative, which causes many students (and some teachers) to refer to the procedure as “deriving”. However, to derive in math refers to a more general process of deducing an equation from certain principles. So there is a need for a more precise verb. I suspect that's why we keep differentiation around. – Matthew Leingang Sep 22 '16 at 13:20
• related: xkcd.com/626 – null Sep 22 '16 at 14:08

## 3 Answers

The derivative (differential) is defined as the limit of the difference quotient

$$f'(b) = \lim_{a \to b} \frac{f(b) - f(a)}{b-a}$$

where difference quotient refers to the difference of $f(b)$ and $f(a)$ in the numerator and the difference $b$ and $a$ in the denominator.

The derivative is also defined (per Leibniz) as the ratio of differentials $dy$ and $dx$,

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

where $dy$ and $dx$ represent infinitesimal changes (differences) in $y$ and $x$, respectively.

As far as history of the term goes, differential was coined by Gottfried Leibniz as described here.

1684 G. Leibniz Acta Eruditorum 3 469 Ex cognito hoc velut Algorithmo, ut ita dicam, calculi hujus, quem voco differentialem, omnes aliae aequationes differentiales inveniti poſſunt per calculem communem, maximaeque & minimae, itemque tangentes haberi

[Just by knowing the algorithm, as I call it, of this method, which I call differential, all other differential equations can be solved by a common method, and maxima and minima, and tangents too, can be found]

Isaac Newton used the notation $\dot{y}$ to denote the generated rate of change in $y$, which he called a fluxion. Leibniz's notations are generally what are used in calculus today, though Newton's dot notation is still sometimes used for derivatives with respect to time, particularly in physics.

• This "difference quotient" suggestion may be an anachronism, in that I think the term "differentiation" was used long before the modern approach of limits and difference quotients (rather than infinitesimals) was invented. – Mitchell Spector Sep 22 '16 at 4:04
• @MitchellSpector That's definitely a possibility. I'll do a little more digging on the history of the term. – Alexis Olson Sep 22 '16 at 4:08
• Very good answer. Would you know where the prime notation for derivative comes from? I've been told it's also Newton's. – Matthew Leingang Sep 22 '16 at 13:14
• Nice answer! Also keep in mind the fundamental principle "differentiation means linearization", since - in school - the differential quotient is commonly established as the solution of finding the slope/incline of the tangent at a certain point $x_0$ of a real-valued function $f$, and the tangent is the linear approximation of $f$ in a neighborhood of $x_0$. – ComplexFlo Sep 22 '16 at 13:42
• @JeppeStigNielsen Lagrange was certainly aware of Newton's notation, and I think even used it in his work at some point. The closest reference I can find is p. 230 of A History of Mathematica Notations: Vol II which states "the Lagrangian $f', f'', f'''$ bears close resemblance to the Newtonian $\dot{x},\ddot{x}, \dddot{x}$, the stroke again replacing the dot. We shall point out that certain English and American writers shifted the Newtonian dots to the position where exponents are placed". – Alexis Olson Sep 22 '16 at 15:35

The etymological root of "differentiation" is "difference", based on the idea that $dx$ and $dy$ are infinitesimal differences.

If I recall correctly, this usage goes back to Leibniz; Newton used the term "fluxion" instead.

According to the Collins English Dictionary, if you "differentiate" between things or if you "differentiate" one thing from another, you recognize or show the difference between them. For a random example, one can say that people have the ability to "differentiate" one person from another.

Although the English definition of the word is different from the Calculus definition, I think that they share the same idea. This is why I think they do:

The derivative of a function f'(x) takes as input any point x that exists within f(x) and outputs the slope of the tangent line at x (the rate of change at x, the rise/run at x). Remember that a derivative is a function and a slope is a number.

Since all curves are made of infinitely many points and since every one of them has its own tangent line with the slope of the tangent line, the action of finding its derivative (finding the line equation of the tangent line) and using that tangent line equation to compute ONE slope among infinitely many other slopes that exist for every point on the curve means to take special note of that one slope. You can say that one slope is distinguished, noticed, recognized, set apart, differentiated from all other slopes. Therefore, I think the English definition, in some ways, captures the Calculus procedure of differentiation.

This is simply my personal intuition and there is definitely a possibility of error in my interpretation and would love to hear any comments or criticisms. This is a supplementary answer to the previous ones and I think they better answer the original question than I do.