Difference between $dy$ and $dx$ I've been taught about $dy/dx$ and how it can be split into $\frac{dy}{dx}=\frac{d}{dx}y$. I'm confused as to why this happens. Don't $dy$ and $dx$ both refer to infinitely small changes in their respective variables? In that case, what is different about a $dy$ that allows it to split into $d$ and $y$?
 A: This is just a notational convention.
$\frac d{dx}y$  does not mean anything different than $\frac{dy}{dx}$. When instead of having $y$ we have some expression we want to differentiate and do not wish to introduce some new variable just for the sake of one equation, writing something like $$\frac{d\left(\frac{e^{-2x^2} - 7x}{\log_\pi(x^2-1)}\right)}{dx}$$ is not really convenient to work with. So instead we prefer to write $$\frac d{dx}\left(\frac{e^{-2x^2} - 7x}{\log_\pi(x^2-1)}\right)$$

Admittedly, there is a somewhat different concept involved in the $\frac d{dx}y$ notation as opposed to $\frac{dy}{dx}$. You can think of it as the operator $\frac d{dx}$ acting upon the function $y(x)$. But even in this case, the operator $\frac d{dx}$ is an atomic notation, not some conglomeration. It is not "something called $d$" divided by "something called $dx$". It is defined only as a whole.
A: First, let us understand what $dx$ and $dy$ mean.
When we prefix $\Delta$ to a variable, it implies a discrete difference: $$\Delta x = x_2-x_1$$ where $x_2$ and $x_1$ are two values that the variable $x$ can assume. In this case, these two values can have a finite difference. When the two values approach each other (as shown in the limit below), the difference approaches to zero: $$\text{as }{x_2\to x_1},\ \Delta x \to 0$$ However, you must remember that this difference is not exactly zero. To write this we prefix $d$:$$\lim_{x_2\to x_1}\Delta x \equiv dx$$ In other words, operator $d$ is same as operator $\Delta$ under the said limit.
Now, prefixing $d$ to any element (or variable like $x$) operates on the element in order to create an infinitesimally small (or in some approximate methods, very small) difference of values of that variable is considered.
Thus $dx$ means a very small difference on $x$ while $dy$ means a very small difference on $y$.
Prefixing the operator $\dfrac{d}{dx}$ means evaluating a derivative of the element following the operator with respect to x. $\dfrac{d}{dx}y$ means to find the derivative of $y$ with respect to $x$. The notation $\dfrac{dy}{dx}$ is derived from the tangent-slope interpretation of the derivative, that is to take the ratio of the opposite side with the adjacent side. The discrete product (using $\Delta$) provides the slope of the secant. Consider a secant along $x$ to a function $y$:$$ \text{slope of secant} = \dfrac{\Delta y}{\Delta x}$$ To calculate the running slope of the tangent to the function:$$\boxed{\text{slope of tangent} = \lim_{\Delta x \to 0} \dfrac{\Delta y}{\Delta x} = \dfrac{dy}{dx}}$$
Thus, $\dfrac{d}{dx} y$ is the correct way to use the derivative-along-$x$ operator on a function $y$ while $\dfrac{dy}{dx}$ is a notational convention hinting towards the slope interpretation of the derivative operator.
Note: There is no algebra involved here. $y$ is not being multiplied with $\dfrac{d}{dx}$, we are simply switching notation for convenience.
