Why are derivatives specified as d/dx? Is the purpose of the derivative notation d/dx strictly for symbolic manipulation purposes?

I remember being confused when I first saw the notation for derivatives - it looks vaguely like there's some division going on and there are some fancy 'd' characters that are added in... I recall thinking that it was a lot of characters to represent an action with respect to one variable. Of course, once you start moving the dx around it makes a little more sense as to why they exist - but is this the only reason?

Any history lesson or examples  where this notation is helpful or unhelpful is appreciated.
Because of their definition:

Start with a function, calculate the difference in value between two points and divide by the size of the interval between the two. You can represent this as such:

$$\frac{f\left(x_2\right)-f\left(x_1\right)}{x_2-x_1}$$

or

$$\frac{\Delta f\left(x\right)}{\Delta x}$$ 

Where ∆, delta, is the Greek capital D and indicates an interval. Now, take the limit as $\Delta x$ goes to zero, and you have the differential. This is indicated by using a lower case $d$ instead of the $\Delta$.

$$\frac{df\left(x\right)}{dx}$$

Now, if this operation is treated as an operator applied to a function, it is usually represented as

$$\frac{d}{dx}f\left(x\right)$$

Note that (typically in physics), you can also use the letter $\delta$ to indicate very small intervals and in general you would use the symbol $\partial $ to represent partial differentials. They are all variations of the letter $D$.
This page, (which quotes from "Yearning for the Impossible, The Surprising Truths of Mathematics") has:


...It has to be explained that dy/dx is not the ratio of infinitesimal differences dy and dx -- since infinitesimals do not exist -- but is rather a symbol for the limit of the ratio y/x as x tends to zero...