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

What is the derivative of a function? It's measure of the slope, or rate of change of a function. So what is $f'(x)$? it's the limit of the ratio of difference in height (difference in $f(x)$), to the difference in $x$, as the differences go to zero.
If you think of the $d$ in $dx$ as shorthand for "difference", then you immediately see that $df/dx$ is a very suggestive notation for $f'(x)$. 
$\quad\quad\quad\quad\quad\quad\quad$
Now, what about implicit differentiation? Even though $df/dx$ is not a real fraction, just a good notation, pertending it is a fraction leads to the law of implicit differentiation:
$$\frac{df}{dx}=\frac{df}{dy}\frac{dy}{dx}$$
Or in our older notation:
$$f'\left(y(x)\right) \text{(derivative acc. to $x$)}=f'(y) y'(x)$$
Now you see the strength of the $df/dx$ notation - there is no confusion about what variable the function is being derived by.
