Why are derivatives specified as d/dx?

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 $D$.

If you have access to it, the book A History of Mathematical Notations, by Florian Cajori, has a pretty detailed description of the history of notations for derivatives in its second volume.

If you're a physics kind of person, then a good reason to like this notation is that it gives the correct units for the derivative: whatever units $f(x)$ is in, the units for $\frac{d}{dx} f(x)$ are obtained by dividing by the units for $x$.

This is the Leibniz notation, which is based on the ratio of "infinitesimals". $dy$ and $dx$ are, respectively, the infinitesimal increment of the dependent variable $y$ and the infinitesimal increment of the variable $x$.

There are other notations: Newton notation, which puts a dot over the variable name, as in $\dot y$, and Cauchy notation, which uses the operator $D$, as in $D(\sin(x))=\cos(x)$.