I have just learnt implicit differentiation and am trying to understand it using the chain rule to create deeper understand/link to previously-taught concept instead of just understanding it in isolation, almost as if I'm trying to work it out myself. However, I have run into some problems notation-wise.
We know that implicit differentiation makes use of the chain rule to work.
Before learning about implicit differentiation, this was what I learnt about the chain rule:
Example: Given that 
, differentiate y with respect to x.
This is a standard example, which we do not really think of when doing explicit differentiation, because well, there really is no need to.
On the other hand, a genuine understanding of the chain rule is required when learning implicit differentiation.
Supposed I am asked to differentiate $sin(y)$ with respect to $x$.
I know that 
but suppose I wish to write this out fully using chain rule.
Making use of the above method in the initial example, I let $u$ denote the "inner function" i.e. $u=y$.
I could of course now write, $y= sin(u)$ now.
Since $u=y$,
The RHS is the answer we seek, because of course we know
but the LHS is problematic, because of my decision to let $u=y$. I end up with 
being cancelled both sides, giving the ridiculous $1 = cos (y)$.
Where have I gone wrong in terms of notation in trying to "prove" that this link between the chain rule and implicit differentiation? I have followed my initial example exactly and yet something has gone wrong. I know it stems from letting $u=y$, but I'm not sure what I should have done. Would appreciate any assistance.