Problem in trying to link chain rule and implicit differentiation

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.

• If you let $u=y$, $y\neq \sin(u)$. I guess you were thinking of $y$ as the function $\sin(y)$. – Javi May 1 '18 at 9:10
• @Javi So what should I have written instead, say let $u=a$? Where $a$ is an arbitrary variable – Charlz97 May 1 '18 at 9:11
• Put $v =\sin{y}$. By the chain rule, $$\frac{dv}{dx} = \frac{dv}{dy} \frac{dy}{dx} = (\cos{y}) \frac{dy}{dx}.$$ – user539887 May 1 '18 at 9:13
• Please learn to use MathJax to format mathematical expressions instead of pasting pictures of them. Images are neither searchable nor accessible to people using screen readers, nor do they show up in summaries. I have to imagine that it would take much less time to simply type in all of that content than photographing, cropping and uploading all of those images. You can find a tutorial and quick reference here. – amd May 1 '18 at 20:33