What is that dx thingy at the end of an integral

Question

Some of my professors claim it’s the infinitismal section of the graph, and some say it’s just notation. Then my professor said the differential of a function and the derivative of a function are two different things, whatever that means. I’m just all very confused. Plz help.

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Answer 1

You may have come across the idea that you can approximate the derivative (slope) of a graph at a point by drawing a straight line from the point to a very nearby point on the graph and taking the slope of that line. For example, if you want to know the slope of sin(x) at a specific value of x, you can draw a line to a point a small horizontal distance h away from x, and the slope of this line is:

[sin(x + h) – sin(x)] / h

For small values of h, this is a good approximation to the slope of the curve, and the smaller you make h, the better the approximation gets. If you want to know the exact value of the derivative, it’s tempting to try and make h zero, but then you end up with

[sin(x) – sin(x)] / 0,

which is just 0/0, so that’s not much help. It seems like you need to find some way of making h infinitely small, aka “infinitesimal”.

The usual treatment of calculus, called “real analysis”, basically sidesteps this whole “infinitesimal” idea and uses a slightly different approach known as “limits”, which are maybe less intuitively obvious but turn out to be easier to deal with and more widely applicable. But there is an alternative approach known as “nonstandard analysis” which gives a proper definition to infinitesimal numbers and uses them to develop calculus. A “differential” is essentially an infinitesimal change in the value of a variable, and is denoted by, say, dy. A “derivative” is the actual slope of the function and is denoted by something like dy/dx.

Outside the world of nonstandard analysis, dx is just notation and doesn’t mean anything by itself. But nonstandard analysis shows that the intuitive idea of infinitely small changes can actually be made sense of, and that an expression like dy/dx can be interpreted as one infinitely small number divided by another (and an integral can be interpreted as an infinite sum multiplied by an infinitesimal). When introducing students to calculus, some people like to lean into intuitive ideas about infinitesimals, while other people feel that’s unhelpful and teach that dx is just notation.