calculus - How can you explain implicit differentiation?

So I am taking calculus 1 online from a local college (bad idea, but the only thing that fit my schedule).

The professor used the notation $f'(x) =$ for EVERY function up until two weeks ago.
All of the sudden he changed his notation to d/d$x$, d$y$/d$x$ and hasn't been able to explain it clearly enough via videos.

I am trying hard to learn how implicit differentiation works, and I've been putting in six hours per day of studying for two weeks. I just don't get it. I have read and tried the examples on MIT opencourseware, our textbook and calculus for dummies.

What is this d/d$x$ that gets placed everywhere? Where do you place it? Why use it? Same with $\text{d}y/\text{d}x$.

I've had lots of success here, so if someone could show me how to do this i would really appreciate it. This is my last hope :)

Thanks