Implicit differentiation can be very tricky the first time through.
I'll try to explain it the best that I can with your first example
So we have:
Simplify it a little bit:
Now we take derivatives of both sides
Now here's the tricky part. Any time we take the derivative of a multivariable function like this one we consider y to be a function of x. So for the sake of understanding how implicit differentiation works, I'll replace every occurence of y with y(x). However, this isn't what you should be doing in your homework.
Now we use properties of the derivative to make our job easier:
Notice the first term contains only x, so we can take our usual derivative:
However, the other terms contain y(x), so we have to use chain rule:
So whenever we see a y(x) (or rather a y), we take the derivative with respect to why and multiply it by
So using the above argument and chain rule
Now we replace y(x) with y and take derivatives:
And now you just solve algebraically for