I have question regarding to product rule.

We know that derivative of y with respect to x is:

Now I'm trying to understand the proof of product rule:

to simplify notation:

Ok, now to proof:

we know that so we can substitute this with respected to u and v in equation and get:

Now to do some algebra we get:

And we know that

so finally we substitute u' and v' to the equation we get:

How i can get rid of since this is some error in my proof.