The solution to some implicit equations might be multiple functions, like in the example in your question:
[tex]x^2+y^2=1\\y =\pm \sqrt{ 1-x^2}[/tex]
and so you have to treat them separately when finding the derivative (this can be much more messy than in the example above)
Instead, it is more efficient to do the differentiation implicitly - once. Lmk if you have questions.
