Have you learned Taylor series? A smooth function can be expanded in a power series around a point a as
f(x) = f(a) + f'(a)(x - a) + f''(a)(x-a)2/2! + ...
See here for more details. Now you know a derivative gives you the slope -- i.e. a straight line. This means you keep only up to first order as an approximation
f(x) = f(a) + (Δf/Δx) (x - a)
Δf/Δx = [f(x) - f(a)]/(x - a)
This is the familiar result for the slope of a straight line. However, this is a finite difference and an approximation. We want the exact result, so we must take the limit to zero. We create a new notation df to mean when Δf → 0, and dx to mean Δx → 0. Hence we get
df/dx = lim (x → a) [f(x) - f(a)] / (x - a)
This is the definition of the derivative.
As for the algebraic manipulations done on them, it's a slight abuse of notation. That's what a mathematician will tell you. I'm a physicist though, so I view it differently. I view it as moving one step back from the derivative to a finite difference, i.e. working with Δf/Δx instead of df/dx, and then performing manipulations with them.