Say $f(x)=x^2$. Sometimes I have seen written
$$
\frac{df(x)}{dx}=2x
$$
Other times I have seen
$$
\frac{df}{dx}=2x
$$
Is there any difference between these two notations? The second one in particular looks odd to me as $f$ refers to a function, rather than the output of a function. To make matters worse, sometimes I see
$$
\frac{df}{dx}(x)=2x
$$
Which seems to imply that $df/dx$ is a function, and $\frac{df}{dx}(x)$ refers to the function evaluated at the point $x$.
Best Answer
They are the same, but I think there's a little bit more to be said. This is not a technical answer, but is how I like to think about it:
${\frac{df}{dx}(x)}$ means "take the derivative of $f$, and evaluate this new function at the point $x$".
${\frac{df(x)}{dx}}$ means "take $f$ at the point $x$ and find the derivative there"
${\frac{df}{dx}}$ is pretty much the same as the first one, but we don't have an evaluation point specified. By convention, we assume we are evaluating at the point $x$. But this doesn't have to be the case. For example, you could do ${\frac{df}{dx}(x^2)}$, meaning "take the derivative of the function $f$ and evaluate it at the point ${x^2}$.
Anyways - overall you can see how they are all essentially equivalent.