How is the modern definition of derivatives different from that of Leibniz' “infinitesimal”-based system?

I pose this question knowing fully well a somewhat similar question was asked in the popular question Is $\frac{dy}{dx}$ not a ratio?

However, as great as the answers are, the answers stop short of explaining a particular culprit I run into every time I try to understand the difference between the modern definition, and that of Leibniz. As such, at least for my own sake and possibly that of others, I feel the need to ask this question.

Derivative picture

The problem I'm having is that when I look at the page on derivatives in my calculus book, I see the all-familiar drawing detailing how to think about the limit definition of derivatives, as pictured above. This is supposedly different from Leibniz' idea of the ratio of two infinitesimal quantities - but I don't understand how.

In both cases, we have a Δy being divided by a Δx. In Leibniz' vision, Δy becomes dy and Δx becomes dx, two "infinitesimally small quantities", smaller than anything imaginable but still greater than zero. In the modern limit definition, Δy becomes dy, an unimaginably small quantity, and Δx becomes dx, again an unimaginably small quantity. That, to my untrained perspective, looks identical to Leibniz' idea of derivatives, except the concept of dy and dx being "infinitely small" is now embodied in the limit.

How is this modern definition any different from Leibniz' definition that relies on the "ghosts of departed quantities"?

Edit: I should emphasize that I'm not looking for answers through the lens of non-standard analysis. This is a regular, standard calculus question.

• You might want to read about hyperreal numbers (which is a rigorous framework for the type of calculus Newton and Leibniz used) and compare with the modern definition of the derivative in terms of a limit which definition is precise (so-called $\epsilon$-$\delta$ definition) and wasn't used by Leibniz contemporaries. – Guest Jan 11 '17 at 6:58
• I'm not too familiar with the specifics of Leibniz's work, though I know he was a proponent of infinitesimals. Newton, to the best of my knowledge, did not use infinitesimals. He understood that he had no good answer for what he meant by the "ultimate ratio" of delY/delX when both become zero. Berkeley's famous "ghosts of departed quantities." But Newton never used formal infinitesimals or attempted to reason with them. In his work his thinking was along the lines of the modern theory of limits, even though he did not have the formalism. This is my understanding. – user4894 Jan 11 '17 at 7:57
• @user4894, what you write about Newton is incorrect. Newton used several approaches to the calculus, one of them based on infinitesimals. – Mikhail Katz Jul 13 '17 at 13:07
• The key difference is that the modern definition uses only precisely defined terms, whereas the definition that Berkeley was opposed to used the term "infinitesimal" which had not been precisely defined. Notice that no real number can be described as "infinitesimal", so if we refer to infinitesimals we are either speaking loosely and imprecisely or else we have left the real number system (and so there is then a great burden to explain exactly what we are talking about). – littleO Jul 13 '17 at 13:25
• The statement "In the modern limit definition, $\Delta y$ becomes $dy$, an unimaginably small quantity" is not correct, and I believe it is a misunderstanding of the modern definition. In the modern definition, there is no such thing as "$dy$" at all, there is only $y'(x)$ which is defined to be $\lim_{h \to 0} \frac{y(x + h) - y(x)}{h}$. We may use the notation $\frac{dy}{dx}$ instead of $y'$, but at no point do we introduce $dy$ or $dx$ as separate, individual entities. Certainly the modern definition makes no reference to "unimaginably small" quantities. – littleO Jul 13 '17 at 13:30

You are correct to find a close connection between Leibniz's differential ratio $\frac{dy}{dx}$ and modern concept of derivative via limits. The connection can be made precise in terms of the shadow. Namely, the limit of $\frac{\Delta y}{\Delta x}$ is the shadow of the ratio $\frac{\Delta y}{\Delta x}$ when $\Delta x$ is infinitesimal. More precisely, we say that the limit exists if and only if the shadow exists and is independent of the infinitesimal $\Delta x\not=0$ chosen.