So now my question is why, if

is so different from

, is it possible to snap components on and off so easily when doing implicit differentiation?

Consider

When we differentiate with respect to x, we can write

And since we don't know how to calculate

, we differentiate that element with respect to y and then tack on (dare I say "apply"?)

after to get

. And I have it stuck in my mind that we do that bit of dy cancellation to convince ourselves that what we've really found is

. Is that what's going on? Are we free to cancel components of two dissimilar notational items like that?

However, I am confused about why you multiplied (is that the right word?) 2x by

. I agree it probably won't do anybody any harm to tack on

to the right of

, but is that actually going on when you compute

? Is it a real step that gets omitted because it's so obvious?