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Newton/Leibniz invented calculus on approximately 1680's. Cauchy/Weierstrass defined the $\epsilon - \delta$ definition of a limit in approximately 1820's.

So how did they define derivatives?

I know that they looked $\frac{f(a+d) - f(a)}{d}$ for $d$ "a very small number", i.e, a number so that $d^2 = 0$ but $d\neq 0$: "infinitesimals".

They defined $f'(a)= \frac{f(a+d) - f(a)}{d}$, and under this definition, the derivative of $f(x)=3x$ is $\frac{3(a+d)-3a}{d}= \frac{3a+3d-3a}{d}= \frac{3d}{d}= 3$.

Suppose $d^2=0$. Then $d^{-1}(d^2)=d^{-1}0 \implies d=0$. So a real number with this property is zero. So this approach is incompatible with real numbers as we usually define them (the axioms of real numbers).

How does this approach work with a different idea of "numbers"?

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    $\begingroup$ They defined derivatives quite differently. Cauchy and Weierstrass already used limits, if I am not mistaken. This is quite far from Newton and Leibniz, only nowadays we know how to formalize infinitesimals. This book might interest you. $\endgroup$ – Ian Mateus Dec 31 '13 at 21:59
  • $\begingroup$ @Thanks. I'll look into it. $\endgroup$ – user87274 Dec 31 '13 at 22:11
  • $\begingroup$ @IanMateus You got here first; do you feel like expanding that comment into an answer with an overview of the idea? If not, I (or someone else, of course) could take a crack at it. $\endgroup$ – Mark S. Dec 31 '13 at 23:03
  • $\begingroup$ @MarkS. I am quite lazy today, feel free to answer it (as everyone else). $\endgroup$ – Ian Mateus Dec 31 '13 at 23:13
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This has been formalized in "nonstandard analysis", which uses "Hyperreals" instead of just real numbers. Hyperreals include positive infinitesimals that are positive but smaller than every positive real number, and they have infinities that are positive but bigger than every real number, etc. There's a lot to be said about the construction of the hyperreals, and why a mathematician would want to use this nonstandard framework, and Terence Tao has already said a lot of it well (not recommended for someone without at least good backing in undergrad real analysis). But how derivatives work once you have the hyperreals isn't so complex.

Every function you're likely to come across in calculus that's defined on an interval, automatically has a "hyper" version defined on the hyper version of the interval (throw in infinitesimal shifts and infinities if the interval is unbounded). Basically, you can input hyperreals into the functions you're used to dealing with without a problem. Also, every hyperreal that's finite (less than some integer in absolute value, let's say) has a nearest real number.

Now, we can take $\frac{f(a+h)-f(a)}h$ for some infinitesimal $h$ (either positive or negative), and we'll get some hyperreal number, which will be closest to some real number (depending on $h$), which I'll call $f'_h(a)$. If $f'_h(a)$ is the same real number for every infinitesimal $h$, then we call that real number $f'(a)$.

For an example, consider $f(x)=|x|$. This is $x$ for real $x\ge0$ and $-x$ otherwise. Its hyperreal version is $x$ for hyperreal $x\ge0$ and $-x$ otherwise. Then $f'_h(0)=h/h=1$ if $h$ is a positive infinitesimal, and $-1$ if it's a negative infinitesimal, so $f'(0)$ doesn't exist.

As Ian Mateus already pointed out, Elementary Calculus: An Infinitesimal Approach is a free Calculus textbook based on this sort of thing.

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  • $\begingroup$ It's worth noting that $h^2=0\Rightarrow h=0$ even when $h$ can take on hyperreal values. The square of a positive infinitesimal is a smaller one, just like the square of any real number in (0,1) is. $\endgroup$ – Mark S. Jan 1 '14 at 0:06
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    $\begingroup$ It is worth noting also that SIA (smooth infinitesimal analysis) does provide a framework with nilsquare infinitesimals. $\endgroup$ – Mikhail Katz Jan 1 '14 at 8:44
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Concerning the OP's comment to the effect that "Cauchy/Weierstrass defined the $\epsilon - \delta$ definition of a limit in approximately 1820's" it should be kept in mind that Cauchy never gave an $\epsilon, \delta$ definition of limit. Thus, starting in 1821 he consistently defined continuity of $f$ in terms of infinitesimal $x$-increment implying an infinitesimal change in $y$; see Cauchy's Cours d'Analyse.

Weierstrass was born in 1815, so that in 1820 he was five years old. He did get around to working on $\epsilon, \delta$ fifty years later, namely around 1870, but Cauchy was long dead by then.

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