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:

Remember:

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