Is there an intuitive way to think about implicit differentiation?

Is there an intuitive way to think about implicit differentiation?
If we differentiate x with respect to itself, then there's no problem. x will change with respect to itself.

But if we differentiate y with respect to x, then we need to add in some extra notation because we are saying y is changing with respect to something else.

Is there an intuitive way to think about this without complex jargon or notation?

put a dy/dx next to every term
derive every term
if its an x term, get rid of the dy/dx
solve for dy/dx

Alright, that's good for becoming a derivative robot.

I'm trying to actually understand what I'm doing.
Give me a real-world example

I mean implicit differentiation is just an application of the chain rule. I mean you can just apply the definition of the derivative to prove the chain rule and you will see it 'works'. Not sure why you need more intuition than that.

implicit differentiation is actually the implicit function theorem, which isn't trivial at all. I dunno OP. it just werks.

>I mean implicit differentiation is just an application of the chain rule.

Yeah, I hear this a lot but for some reason I can't wrap my head around it.

What? No. How much it fucking takes to define properly what a function is and use proper notation and state "DIFERENTIALS ARE MERELY MNEMONIC AND SHOULD BE BE NOT TAKEN LITERALLY".

it's not. you're being misinformed.

thinking in terms of "x is changing with respect to y" is dangerous and might confuse you when learning about the ("total") derivative in multivariable calculus, but it might be the way to go...

all it takes for your silly mnemonic to break is a function in more than one variable

well, it's not completely wrong. let me elaborate.

the implicit function theorem tells you that if you have a function f(x,y), then, under some conditions, you can "write y as a function of x", that is, write f as a function f(x,g(x)). therefore what you call "y" is just a function g(x), and it makes sense to use the chain rule to differentiate f(x,g(x)).

dy^2/dx = yx or some shit right? like an x appears out of nowhere

Alright so could you give me an intuitive example to have a good conception of implicit differentiation?

that only works for seperable DEs though right? but you can still use implicit differentiation on inseparable DEs too

Yeah lol

It hurts my brain because it's all abstract to me. I need concrete, tangible examples to understand something.
When it's just notation and no explanation it's confusing

let's take the circle, that is, the relation
x^2 + y^2 = 1
which is just the inverse image at 1 of the function f(x,y) = x^2 + y^2
now we are interested in the slope of the tangent line, where it makes sense. so thanks to the implicit function theorem, under some conditions we can write y as y(x), a function of x. therefore we can differentiate:
f(x,y(x))' = (x^2)' + (y(x)^2)' = 2x + 2y(x) y'(x)
now since f is constant in the circle, f(x,y(x))' = 0, and so
2x + 2y(x) y'(x) = 0
implies
y'(x) = -x / y(x)
which means the slope of the tangent line at the points of the circle where this construction makes sense is just -x/y. right away you can see that if y=0 this isn't even defined, it turns out those are the points where it doesn't make sense.

that's a general result for the inverse image at regular values of differentiable functions. manipulating meaningless mnemonics in DE classes is usual though.

Now it's starting to make some sense.

You mentioned that in a circle, for instance, certain values of x allow us to write y=y(x), right?

So does that mean that for other values of x, this wouldn't work? What if I tried to differentiate on those values?

well, look at the circle. what happens near the point (1,0)? can you write that part as a function of x? if you can't, then you can't even define what differentiating "y" means there

holy fuck, this is why I got a C in calc II

Don't listen to people who say "it just works". For me, it took me a while to really understand implicit differentiating, it's hard for me to explain since I think of it a different way and I feel like I would just over complicate it. I'd recommend to just really try to think about it

Take Real Analysis.

Sorry if that's bad advice, but it's what I had to do

So, around the points where the derivative you obtain is not defined, it means that Y does not change according to changes in X, which means you cannot write Y as a function of X because oh that point I does not depend on x (when, for example on the circle, Y is 0 and the value of X depends on the radius, So x can be whatever -the value of the radius - and Y will be 0, thus not being a function of X).

In real life, think it like... correlations... Like, you not having money is due to a drinking problem and having lots of friends, one could say and modelate it assuming you drink too much because you have lots of friends (Y is a function of X), but that is not true if none of your friends drink or all your drinking friends die of liver failure (y cannot be expressed as a function of X in these cases). It's the best example I can think of, applications in real life do apply the existence of correlations (it is implicit in the fact that Y can be written as a function of X, they must be correlated somehow), but I can't think of something you can go an read to understand it. Also, excuse my poor English, I literally learned this shit this year so I must not have the best grammar. I hope I could convey something in my post.