I'm trying to understand the intuition andprecise meaningof each component of common differentiation notation.

As I understand it:

$\displaystyle

\frac{dy}{dx} = f'(x) = y'

$

Each form of notation asks "for our function of x, what is the slope at point x?" Or, in the case of $\displaystyle

\frac{dy}{dx}

$, "how much of 'little bit of y' is there for every 'little bit of x'?" (which is the same as asking what the slope is). So far so good. But suppose we have a basic function $\displaystyle

y = x^2

$. What is the correct $\displaystyle

\frac{dy}{dx}

$ notation for this derivative, andwhat is the specific meaning of each part?

Is it:

$\displaystyle

\frac{dy}{dx}y=\frac{dy}{dx}x^2

$

The right side makes sense ("how much change in y is there for every change in x when we consider the function $\displaystyle

x^2

$?"), but the left side doesn't seem to. Can you "ask" $\displaystyle

\frac{dy}{dx}

$ of y? How do we get just $\displaystyle

\frac{dy}{dx}

$ instead of $\displaystyle

\frac{dy}{dx}y

$?

Alternatively, we could do:

$\displaystyle

\frac{d}{dx}y=\frac{d}{dx}x^2

$

Now the left side makes sense. We want to solve for $\displaystyle

\frac{dy}{dx}

$, and placing just a $\displaystyle

\frac{d}{dx}

$ next to the y allowed the y that was already there to complete the notation. But the right side doesn't make sense anymore. What is the numerator "asking"? If we allowed x to complete the question like we did on the left side, we would get something like $\displaystyle

\frac{dy}{dx}=\frac{dx}{dx}x

$, but that is almost certainly incorrect.

Or maybe I didn't get it right with either option I offered up. Let me know. And whatever it is, please specify what the notation '"is asking" in each component.