Where did I go wrong?

[zenith wraith]

Suppose

A = f(B)

B = g(C).

Therefore, A = f(g(C)). (Right?)

Therefore, A' = f'(g(C))•g'(C).

Therefore, f'(g(C))•g'(C) = f'(B)•B'.

Therefore, f'(B)•B' = A'•B'.

This means that A'•B' = A', implying that A' is a function of itself??

Is this possible? Where did I go wrong?

[lavish verge]

A' ≠ f'(B)

A' = f'(B)•B' is correct

One thing is the derivative of f(g(x)), and a different thing is the derivative of f evaluated at g(x): f'(g(x))

$\left(f(g(x))\right)'\not= f'(g(x))$

[silk mulch]

ELeonardo

[zenith wraith]

Wouldn't this be incorrect, too, though, since B' ≠ g'(C)?

[lavish verge]

I assumed you're taking derivative with respect to C 🤔

If C is a function of x and you're differentiating with respect to x, then the complete chain rule is
A'=f'(g(C))•g'(C)•C'

[zenith wraith]

Hmm I'm honestly not too sure myself. This question arose from an application concept. My scenario is envisioning that A is influenced by B, B is influenced by C, and thus A is influenced by C. I'm thinking of applications, so it could be multi-variable, right? Or do the number of variables matter?

[zenith wraith]

I kind of didn't think about whether C might be a function. 😅 Since I was thinking in terms of applications, I just said C was C without thinking about it. My point above addresses this though, I think

[lavish verge]

Oh ok. Well one thing is composition of functions, and a different thing is a multivariable function

If A is influenced by both B and C, and they are independent, then A is a multivariable function
A=f(B,C)

[zenith wraith]

Of course, but I'm wondering if there is a generalized form without specifying the variables

[zenith wraith]

This makes sense to me

Would you assume anything about C though?

[lavish verge]

I was considering it to be a variable now, as well as B

[lavish verge]

Generalized form of the chain rule?

For n compositions?

Or do you mean a generalized form of the derivative for multivariable functions (?

[zenith wraith]

I suppose. Without variable specifics though, you're right, it doesn't seem possible to make a general form

[zenith wraith]

No, I'm familiar with what to do in that situation. This is perhaps a philosophical question of what to do with functions you don't have a lot of information on

[lavish verge]

Ohh well I guess it all depends on what you're differentiating respect to. I mean you can write anything on paper, any derivative as long as it follows the chain rule. You could use a big Product for the notation. The same for partial derivatives of multivariable functions. But if you don't know the specific relations, you won't get any numerical value you want

Mmm

In dynamic systems, everything that's not really identified is called disturbance

[zenith wraith]

I think that it wouldn't be a disturbance, solely because I'm not completely ignorant to the functions I'm considering. From what I understand about dynamics, which is not very much, it's about finding the actual function and using a process that is usually based on data. I'm not really so concerned with exact function or numerical value(s), so much as how general ideas would change (ie differentiate); basically, proportionality and what changes affect others. That's why my brain didn't go to function variables. I think I'm talking less about rigorous equations and more about broad ideas