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While I was looking at a question regarding derivatives, I suddenly got enlightened when I realized, $dx=\lim_{h\rightarrow 0} (x+h)-(x)$

I noticed this while considering on the equation $\frac{df(x)}{dx}=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-(x)}$

Up until now, i am surprised that I've never realized this before because now every differential equation and integrals make sense now.

For example, dx at the end of the integral is actually meaning the width of the rectangle over the x axis.

And also, I think it is much easier to grasp the differential equations or integral when it is shown like this.

I wonder, is there anything wrong about this equation, if not, why isn't it taught in this way, Instead differentials and integrals are treated entirely different things, and we just try to deal with "dx" by not having a single idea about what that actually means.

Gary
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Emin Bedir
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    See also https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio and https://math.stackexchange.com/questions/1906241/when-not-to-treat-dy-dx-as-a-fraction-in-single-variable-calculus and https://math.stackexchange.com/questions/2525211/why-do-people-say-dy-dx-is-not-a-fraction-but-then-use-it-as-one-when-doing-the and https://math.stackexchange.com/questions/3819116/rigorously-whats-happening-when-i-treat-fracdydx-as-a-fraction for related discussions. – Arthur May 28 '24 at 08:53
  • Also https://math.stackexchange.com/q/3775923/26369 and probably many others – Mark S. May 29 '24 at 10:21
  • We think about "$\mathrm{d}x$" as "a very very small change in $x$". But "$\lim_{h\to 0} (x+h)-x$" is just zero. Not just some very small number, exactly zero - that's what the limit does. So your idea is a great way of informally thinking about it! But it's not fully accurate to how we formalize the idea mathematically. – Deusovi May 29 '24 at 19:53

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When you write an equation of the sort "$dx=\lim_{h\rightarrow 0} (x+h)-(x)$", you are not using the limit notation in a traditional way, so most users here will disapprove. What you seem to mean is that when $h$ "becomes" infinitesimal, it is helpful to denote it $dx$.

On the other hand, when you write the formula $\frac{df(x)}{dx}=\lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-(x)}$, you are using the traditional meaning of the limit, and the formula is readily understandable to users here.

While I should point out that you should not use the same notation for two distinct ideas, I sympathize with the thrust of your observation to the effect that if one thinks of $h$ and $dx$ as being infinitesimal, it helps understand both the concept of derivative and the concept of integral (thinking of $dx$ is being the infinitesimal width of the vertical rectangle).

This point of view is developed fully in Keisler's textbook on calculus with infinitesimals.

J. W. Tanner
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Mikhail Katz
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