By 'prime' I'm assuming you mean the prime symbol, like f -> f' -> f'' -> f(3) -> etc.
Don't get too used to that notation, the dy/dx notation is preferred because it works better in so many cases. It also makes nth derivative much easier:
y -> dy/dx -> d²y/dx² -> d³y/dx³ -> etc...
It also makes separation of parts (something you'll see later) much easier to comprehend - it 'looks' like you multiply both sides by dx, but the "d/dx" is one symbol and *technically* you can't split it, but it makes so much sense to think of it that way (it was actually designed that way). You also have to use the notation for integrals (the opposite of derivatives).
Well, you could build the definition of the nth derivative (by piling the definition of derivative on top of itself n times) and you can find a pattern in derivatives of most simple functions, but in general, there is no way to get the nth derivative, other than taking the derivative n times. If you don't believe me, try finding a pattern in derivatives of tan(x).
...it 'looks' like you multiply both sides by dx, but the "d/dx" is one symbol and *technically* you can't split it, but it makes so much sense to think of it that way (it was actually designed that way)....
d/dx is the differential operator and is considered one "thing," it's true. When the operator is apllied to a function, however, it represents the ratio of the (infinitesimal) changes of the dependent variable to the independent variable. It is this "duality" that makes many amazing things possible. For example, the derivative of the arcsin function.
Let y = arcsin x
then x = sin y
d/dx[arcsin x] = dy/dx = 1/ (dx/dy) = 1/ (d/dy[sin y]) = 1/ cos y
Then you just plug in and simplify. It's more writing than math - that's the thing about math, when you understand it, your only challenge is tedium with some problems. Don't even get me started on Rotation of Axes...