The derivative has four common notations and an exam uses all of them. $f'(x)$, f prime of x, names the derivative as a function; $f'(3)$ is that function at 3, a plain number, the slope at $x = 3$. If the function is written $y = f(x)$, the same derivative is $y'$, y prime. Leibniz's notation is $\dfrac{dy}{dx}$, read aloud dee y by dee x, meaning the rate of change of $y$ with respect to $x$: $y$ is the quantity that responds, $x$ the quantity that changes, and the $d$'s mark changes that have shrunk to nothing, so the symbol remembers that a derivative began as a ratio $\dfrac{\Delta y}{\Delta x}$ (delta y over delta x, a change in $y$ over a change in $x$). It is one symbol, not a fraction with a number $d$ in it, and it is not $y$ divided by $x$.

The fourth form, $\dfrac{d}{dx}$, read aloud dee by dee x, is an instruction with a blank: $\dfrac{d}{dx}\left(x^3\right)$ means differentiate $x^3$ with respect to $x$, and the answer, $3x^2$, is what the instruction produces. To name the derivative at one point in Leibniz's notation, write $\left.\dfrac{dy}{dx}\right|_{x=3}$, dee y by dee x evaluated at x equals three, the same number as $f'(3)$. With respect to matters: $\dfrac{dV}{dt}$ is how fast a volume $V$ changes as time $t$ passes, and $\dfrac{dV}{dr}$ is how fast the same $V$ changes as a radius $r$ grows, two different derivatives of one quantity.