Thread: Can somebody help explain dx to me?

1. Can somebody help explain dx to me?

I still don't get it, my calculuss BC teacher is not very good at what she does. Just what is dx, why am i incorporating it into every one of my antiderivative, why is delta x part of the integral for area integration, and just why do i use it for. When i convert it back to the original function it just dissapears, so what's the point?

2. Originally Posted by insanity999
I still don't get it, my calculuss BC teacher is not very good at what she does. Just what is dx, why am i incorporating it into every one of my antiderivative, why is delta x part of the integral for area integration, and just why do i use it for. When i convert it back to the original function it just dissapears, so what's the point?
If you have

$\int{f(x)\,dx}$

it is saying "Find the integral of $f(x)$ with respect to x".

The $dx$ also acts as the "full stop" of the integral. So it's saying "Find the integral of everything from the integral symbol to here".

This is similar to the Leibnitz notation for the derivative...

$\frac{d}{dx}\left[f(x)\right]$

which is saying "find the derivative of $f(x)$ with respect to x".

You will find when you get into harder integral techniques (substitution, integration by parts) that using this $dx$ notation makes for easier manipulations.

3. But the important thing is to be able to blame someone else!

4. Also on a smaller scale, when you are using rectangles to find the area under a curve, you want to get the most precise answer possible. This is where the dx comes in. You want an infinite number of rectangles, so the small-ever so tiny change in x for each rectangle is the dx. dx is like delta x, or change in x. So essentially you want to sum up all the rectangles with width dx, which is where the integral steps in, and gets you the most precise answer.

Hope this helps

5. While in basic calculus it's best in my opinion to just think of dx, dy or whatever as just indicating the variable of integration. It's really easy to make false assumptions about these differentials, especially with derivatives like dy/dx.

So anyway, just think of it as notation that indicates the variable of integration. If you didn't have this info, you wouldn't know if you should integrate with respect to x, y, z, etc.

6. I think of dx as an infinitely small change in x.

For example, When taking a riemann sum, we must add add up all of the approximating rectangles each with width $$\Delta{x}$$, but when we take the limit, $\Delta{x}$ becomes ininitely small so that our approximation turns into an exact value.

Another comes from the definition of the derivative. You know that slope is defined as

$\frac{y_2-y_1}{x_2-x_1}=\frac{\Delta{y}}{\Delta{x}}$.

Well, this gives the slope of a SECANT line. To know the slope of a TANGENT line, the difference between $x_2$ and $x_1$ must be infinitely small. Hence the limit:

$\lim_{\Delta{x}\to0}\frac{\Delta{y}}{\Delta{x}}=\f rac{dy}{dx}$

I hope this helps you.

Liebnitzs' notation by far makes the most sense. Think about it for a while, it'll reveal itself.

7. I'm glad you posted that VonNemo19 because this is the issue I was referring to.

From my understanding, "dx" can mean a number of things in different contexts and there is dispute with these as well.

"dx" is first used with limits, like you said, and has the idea of meaning instantaneous change as well as an infinitely small value. When showing the notation dy/dx for a derivative though, I have not seen any textbooks do more than just show that this notation is convention. What I mean is that the idea of dy and dx being independent things is not addressed and they only have meaning when grouped together this way.

Later on the chain rule gets discussed and it seems that these work like fractions and can cancel with normal algebra rules. This isn't always true though and there is a semi-famous example where the result of three derivatives multiplied is -1, when it appears to be 1 by cancellation.

Then with integrals we see dx again, but it seems to be simply notation. Having the dx on the end of every integral doesn't literally mean multiply as it would normally. The same idea of it is the same, but it's derived in a different way than the derivative is.

Then you have differentials like $dy=\frac{dy}{dx}dy$ where again normal algebra rules seem to apply. Differential equations separate variables in the same way, but if you use alternate notation this problem doesn't arise. $ydy=x^2dx$ for example could be written as $(y)'=(x^2)'$.

Then apparently in advanced geometry these differentials are used much more rigorously, but I don't know enough to comment.

My main point is that the notation dx and dy, and others, are presented as notation for specific things but they aren't rigorously defined and it's a huge assumption to apply all algebra rules to them. Often it doesn't matter, but doing so is missing concepts in my opinion.

I have been reading about this issue for a little while but am not saying I am 100% right on everything. I would love to hear from our more advanced posters on this issue.