Why are there so many notations for differentiation? There are so many notations for differentiation. Some of them are:
$$
f^\prime(x) \qquad \frac{d}{dx}(f(x))\qquad \frac{dy}{dx}\qquad \frac{df}{dx}\qquad
D f(x)\qquad y^\prime\qquad D_x f(x)
$$
Why are there so many ways to say "the derivative of $f(x)$"? Is there a specific use for each notation? What is the difference between $\dfrac{d}{dx}$ and $\dfrac{dy}{dx}$? I am only asking this because I am worried that I might use the wrong notation sometimes. For example, I don't know when I should use $\dfrac{dy}{dx}$ instead of $D_xf(x)$, or vice versa. I thank you in advance for your answers.
 A: For the most part, the things you've written are equivalent, and the reason there are so many is partly historical, partly practical (e.g. $D_x$ is better notation when one is using the language of operators or partial derivatives, $y'$ saves space when it's unambiguous, etc.).
But there are two really huge points here.  First of all, the difference between $\frac{df}{dx}$ and $\frac{dy}{dx}$.  It's very important not to confuse these—put simply: $f$ and $y$ are different letters!  They might mean the same thing in certain problems, but they might not—e.g. $y=f(x)$ is a curve in $\mathbb{R}^2$, where $f$ and $y$ mean mostly the same thing, $z=f(x,y)$ is a surface in $\mathbb{R}^3$, where $f$ and $y$ mean totally different things.  Usually, $f$ denotes a function $f(x)$, and $y$ denotes a coordinate, but you should always look at the notation of the specific problem before you make assumptions—you should be able to handle a question about $(q,w)$-plane instead of the $(x,y)$ plane, without getting confused.
The other point is the difference between $\frac{d}{dx}$ and $\frac{df}{dx}$.  It's also very important not to confuse these!  $\frac{df}{dx}$ is the derivative of $f$ with respect to $x$, and it's a function of $x$.  $\frac{d}{dx}$ is just the derivative with respect to $x$, and it's not a function at all—it eats functions and spits out their derivatives: $\frac{d}{dx} (x^3+3x) = 3x^2+3$, $\frac{d}{dx} (e^y+f(x)) = e^y \frac{dy}{dx} + f'(x)$, and so on.  When we write $\frac{dy}{dx}$, we just mean $\frac{d}{dx} (y)$.  Keep these concepts separate—this is the same as the comparison $+1$ versus $y+1$, or $\sqrt{\phantom{1}}$ versus $\sqrt{y}$.
A: $f'(x)$ is equivalent to $\frac{d}{dx}(f(x))$. The difference is that in the first you aren't making explicit that you are differentiating with respect to $x$, while in the second that distinction is made clear. Although when we write $f'(x)$ is usually implied that the differentiation is with respect to $x$. $\frac{df}{dx}$ is also the same thing, in more compact notation. 
$\dfrac{dy}{dx}$ and $y'$ are the same, but this time differentiation of $y$. It is not related to $f'(x)$ in any way, unless of course you have a relation in $y$ and $f(x)$.
I have not come across the notations $Df(x)$ and $D_xf(x)$ so cannot comment on that.
A: A short answer is that in calculus you do lots of symbolic manipulation, so different notations are worth the bother to minimize eye sore and give you what power you need. For instance, the fraction notation helps if you are doing cancellations or partial derivatives.
