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How to find $F_y(y+dy)'$?

Particularly I'm trying to learn how to work with Random Variables. But this question is on pure differentials.

I know there is

$f(g(x))'=f'(g)g'(x)$,

but what is $(y+dy)'=$?

I guess $(y+dy)'=0+something$. But what is that "something" and how to find it?

I'm poor on knowledge with differential algbera, but I'm trying to build some knowledge base that will allow me to solve problems on my own in the future.

Ideally I'd expect the result to be of a form:

$F_y(y+dy)'=F_y'(y+dy)(0+something)$

where $y'=y_0'=0$.

I guess (it's a guess too), that dy' is 0 too, but it looks suspicious as if to me. Because if $y$ is some constant number, then I think dy could make "a small" but still a difference on $F_y(y_0+dy)'$.

Vitali Pom
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  • Right now, the question doesn't make sense (because it could mean a couple different things, you are mixing and matching notations in an ambiguous way). When you say random variables, are you working with stochastic calculus? Because there are specific rules for the differential elements there that don't apply elsewhere, which can clear up your question. – Ninad Munshi Oct 02 '20 at 17:46
  • I think I shall remove the random variables tag then :/ Removing it now. – Vitali Pom Oct 02 '20 at 17:49
  • I don't understand the point of confusion I did ^^ but I removed the tag to stay on the safe side. It shall be pure derivatives now (I hope!). – Vitali Pom Oct 02 '20 at 17:51
  • Oh no, I was hoping for the opposite. Are you doing stochastic calculus? The question becomes less ambiguous if that were the case. Otherwise we can try to list the possible interpretations in an answer. The reason its ambiguous is because it is bad form to mix and match the differential and prime notation, both of which can be interpreted as limits. Which limit do you take first? Is it a differential element to be summed or is it a function? Etc – Ninad Munshi Oct 02 '20 at 17:54
  • I edited the question, could you check if it makes sense now? I've been told that I shall take clearance conversations to the chat until I finish editing the question. Do you know how to initiate a chat here? I've not been a power user of stack as you can see. But I'd need to understand what do you refer to by stochastic calculus. – Vitali Pom Oct 02 '20 at 18:03
  • Talk in chat room is continued here (Ninad join :) ) https://chat.stackexchange.com/rooms/113667/discussion-between-vitali-pom-and-ninad-munshi – Vitali Pom Oct 02 '20 at 18:12
  • Ok, I figured out on my own that I wasn’t approaching the problem from the right direction and hence why there are no answers. I was trying to find dF(y). The problem was that I had not realized the meaning of differentials and of dF(y) and in more general terms of df(x) - which is the small difference in y coordinations while we move by a small fraction and do a small difference in the coordinations in x; That we calculate by the coefficient of the tangent at any point on the graph of the function. dF(y) is of course f’(y)dy. This is according to the tangent formula at a given coordinate. – Vitali Pom Oct 02 '20 at 18:51
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    For example, "$f(g(x))'=f'(g)+g'(x)$" is wrong. – GEdgar Oct 02 '20 at 18:59
  • For general knowledge, @GEdgar why doesn't it work here? – Vitali Pom Oct 02 '20 at 19:12
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    $f(g(x))'=f'(g)+g'(x)$ is wrong. It should be product, not sum. – GEdgar Oct 02 '20 at 23:55
  • Oh my goodness! Thank you so much!! – Vitali Pom Oct 03 '20 at 14:05
  • So we don't know how to calculate this this way because we don't know how to work out two different $a_1-a _2$ while a is $a=f'$ ($a$ is coefficient of the tangent)? – Vitali Pom Oct 03 '20 at 14:46

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