Problem

Many people have asked this question, and many will continue to do so. It is the natural question of someone first learning the subject of calculus: what is “\(\mathrm{d}x\)”, and why is it everywhere in calculus?

Frankly, it’s mostly Leibniz’s fault. Leibniz, a brilliant philosopher and mathematician who may or may not have invented calculus depending on whom you ask, introduced the notation. His view of derivatives was as the ratio of related infinitesimals. In slightly more modern terms, $$\lim_{\Delta x\rightarrow 0}\frac{\Delta y}{\Delta x}=\frac{dy}{dx}.$$ Unfortunately, people have carried this view for an unhealthily long amount of time. A branch of analysis known as “nonstandard analysis” found a clever and eponymously nonstandard way to make the idea of the infinitesimal rigorous. Still others have questioned the validity of the law of the excluded middle, which says that a proposition must be either true or false. By not accepting this law, some finagling and rather nonstandard logic can bring about the idea of an infinitesimal. The list goes on. In short, there is a myriad of “nonstandard” ways to realize \(\mathrm{d}x\).

This brings us to a much more refined question: is there a standard way to define \(\mathrm{d}x\)?

We will defend the claim that the answer is a resounding “yes.” What’s more, we will attempt to demonstrate that the concept is intuitive and natural, even to those relatively new to the subject of analysis. more »