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I have studied the oncept of derivatives recently and I enjoy learning it very much, but I feel no need for that sign. What's the reason we need it?

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    By "sign" do you mean the symbol $\lim$ which denotes the limiting process? Also, how do you define $dx$ on the right hand side of your equation? Is that just your symbol for "a small (but finite) change in $x$" or do you literally mean an *infinitesimal* change in $x$? – 2017-02-20
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    Yes. I mean the limit "symbol". I feel I don't need it. – 2017-02-20
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    I mean an infinitesimal change in x because AFAIK my imaginary camera needs to shoot every single subtle variation. – 2017-02-20
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    Ah. Then that's not the usual definition of the derivative and maybe you wouldn't need $\lim$. But then the question becomes *what is an infinitesimal number*? Certainly no real numbers are infinitesimals (as that is forbidden by the Archimedean property of the reals). Where did you see this equation? Are you studying some variety of non-standard analysis where infinitesimals make sense? – 2017-02-20
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    Well I read this article: https://betterexplained.com/articles/calculus-building-intuition-for-the-derivative/and then came up with this equation: – 2017-02-20
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    Can you correct the link? It is incomplete. I think there is an additional "and" at the end that doesn't belong there (?) – 2017-02-20
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    @MPW Remove the "and" from the end of the link after you click on it and it'll work. – 2017-02-20
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    @M-J OK. I skimmed that link. The first half is just the author trying to give you an intuitive view of the formula for the derivative. But be careful where he says things like "Change by the smallest amount possible (dx)". He's trying to be intuitive, but in the context of *standard* analysis, that line is non-sense. The last half of the link is the author trying to intuitively describe limits after possibly confusing you in the first half. – 2017-02-20
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    When he says things like "The derivative is "better division", where you get the speed through the continuum at every instant", he's describing the limiting process applied to **finite** changes. This is (one way to describe) the real (meaning standard) definition of the derivative and why we need the $\lim$ in that definition. – 2017-02-20
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    I suggest if you want to really understand the derivative, just pick up a calculus textbook and start working through it. There are textbooks that take a more intuitive approach (like this [free](http://www.lightandmatter.com/fund/) one) and ones that take a more rigorous approach (see Spivak's *Calculus*). But any will do a better job than that site (which I don't think was really intended to *teach* the derivative but simply to help students who have already been introduced to it understand it *intuitively* a little better). – 2017-02-20
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    Thanks for introducing this interesting book and your explanation. – 2017-02-20
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    No problem. :-) – 2017-02-20
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    You are probably thinking that $dx$ is an "infinitesimally small" number, but if you try to make that idea precise you will run into trouble. For example, I'm sure you would agree that $dx$ is not a real number (there is no such thing as an "infinitesimal" real number). So what is $dx$? To avoid the difficulty in defining $dx$, mathematicians came up with the limit definition of the derivative. – 2017-02-20
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    @littleO What's wrong with unreal numbers like dx? – 2017-02-21

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The limit sign is there because if it wasn't there, you end up with 0/0. So, the limit sign is there to tell that you let it get closer and closer to 0, and see what the whole thing ends up approaching.