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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https://en.wikipedia.org/wiki/Integral#Terminology_and_notation
“The symbol dx, called the differential of the variable x, indicates that the variable of integration is x.”
I’m going to elide over the history of mathematics and the calculus and all of that stuff and just cut to the quick:
Both sets of professors are right. There is a deeper meaning to d/dx where it does actually refer to the actual change in x, but *for your purposes,* you can treat it as notation. You’re integrating what’s between the integral symbol and the d/dx notation.
>Then my professor said the differential of a function and the derivative of a function are two different things, whatever that means.
Your professor is right! The derivative of a function measures *the rate at which the function is changing*, and the differential of the function *measures how much it actually changed.* For most purposes, in most cases, the difference is purely academic, but there are cases where it matters.
> [Why is it useful](https://math.stackexchange.com/questions/23902/what-is-the-practical-difference-between-a-differential-and-a-derivative/23914#23914) to have the distinction? Because sometimes you want to know how something is changing, and sometimes you want to know how much something changed. It’s all nice and good to know the rate of inflation (change in prices over time), but you might sometimes want to know how much more the loaf of bread is now (rather than the rate at which the price is changing). And because being able to manipulate derivatives as if they were quotients can be very useful when dealing with integrals, differential equations, etc, and differentials give us a way of making sure that these manipulations don’t lead us astray (as they sometimes did in the days of infinitesimals).
It’s the opposite of the S sign in the beginning of the integral.
It helps to think of integrals as the sum of an infinite number of infinitely small areas.
The S sign means sum of infinite terms. The dx makes the area tiny. And that’s how integrals are meant to work.
The integral is the area under a function. This can be done by making a lot of rectangles, and taking the sum of the area of each rectangle. (https://en.m.wikipedia.org/wiki/Riemann_sum for some visualisation). The rectangles will have an area of f(x_i) * ∆x, where x_i is a different point on the x-axis (f(x_i) becomes the rectangle’s height), and ∆x is the width of each rectangle.
When the amount of rectangles go to infinity, in order to get the exact area (the infinite series will converge, this is called a riemann sum), the width ∆x will go to 0. When taking the limit as ∆x goes to zero, ∆x becomes dx (this is notation). We also change the notation from being a sum (large sigma) to the integration sign you’re probably familiar with. Thus, the dx is sort of a remnant of the infinite series.
It also tells you which variable to integrate, which is nice.
This is at least how I think of it.
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.
In a usual integral it is just notation. But sometimes you see things like integral(f * dx + g * dy). In this case, dx and dy are differential forms.
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