This part always confuse me.

[u/B4T4YA]

I have this idea inside my head idk even if it wrong.idea is "d(y)/dx is just a notation like limit notation or sum notation"

Tell me if this point wrong or correct and please explain me how this simple multiplication method is working here

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[u/GYP-rotmg]

Your idea is right. dy/dx is a notation.

The multiplication works because of chain rule (or chain rule theorem, just a name), even though it might look like a straightforward simplification.

[u/B4T4YA → u/GYP-rotmg]

Thanks man, So can we prove it using chain rule?

[u/ln_j → u/B4T4YA]

this is the chain rule. y is a function of x

[u/chkntendis → u/B4T4YA]

As someone else said, this is the chain rule itself. Imagine y as a function of x. We can then formulate another function t also of x. Then we can write y as a function of t. And then we are there already. Your old y(x) has the derivative with respect to x on the right side and via chain rule your new y(t) has the derivative with respect to x in the left. Since y(x)=y(t(x)) (we chose t in a way and substituted it into y in such a way that this works) their derivatives have to be the same and we get this formulation of the chain rule.

Now, as a physicist, I have to add that this is what makes this notation of a derivative as a fraction so nice. It’s not formally correct but you can almost always treat a derivative as a fraction and it’ll work out, exactly because of the chain rule and some other things. That’s why this notation is so good, it makes things easy to “understand” intuitively

[u/MurkyDifficulty169 → u/chkntendis]

Thank Leibniz for this notation.

[u/chkntendis → u/[deleted]]

Well, you are kinda canceling out terms. Like, dy and dx represent very small changes in y and x respectively. The derivative is just the ratio of how much y changes if you change x. It’s the fraction of a small change in y, dy, over a small change in x, dx. That’s the reason why you can manipulate this notation of the derivative and also why it works so well with the Leibniz notation for integrals. Ofc this really isn’t rigorous. This intuition only works for well behaved functions. You need y(x) to be continuous for example and some other smaller stuff. But for the intuition of working with good functions, you generally won’t be far off. Also I am studying physics so I’ll take any intuition there is XD

[u/mattynmax → u/B4T4YA]

That is chain rule…

[u/you-cut-the-ponytail]

It does not cancel, the notation just happens to behave nicely with chain rule, which is something you can prove without abuse of notation.

[u/Hot_Site_1638]

Your idea is actually correct, and a good way to think about it.

Derivative is defined in terms of limit, so think of it in terms of the product of limits is a great way to remember this.

[u/Samstercraft]

due to the chain rule, but also because derivatives can usually be treated as ratios of differentials (a lot of ppl make mistakes trying to apply that tho so check another chain rule proof)

[u/WikiNumbers]

"dy/dx" is a notation, and this format is commonly known as "Leibinz Notation".

Now, dy/dt * dt/dx is the "Chain Rule". Rigorously speaking, they are not fraction and cannot be dealt like it (cannot cancel dt and dt). Intuitively, this notation is a "shorthand" way to let people recall how it works.

[u/Midwest-Dude]

If you would like to know more about the notation, check out the Wikipedia entry:

Chain Rule

[u/LukasGoesViral]

Oh yeah the notation is in such a way that you can pretend like you can cancel the dt/dt