I have a question about implicit differentiation.

In the case of differentiating the expression x^3+y^3=6xy the right hand side becomes 6xy'+6y where y is regarded as a function of x.

using the product rule

then

so I get

which is wrong.

Why is the right hand side equal to ?

I (think) I know the key is that y is a function of x but somehow its not clicking as to why the answer is as it is.