Can ∂x and ∂y in a derivate be seen as ∂ times x or ∂ times y? I'm watching some tutorials on machine learning and know just enough calculus to have an intuition on what a derivative is, but that's it. But this question is bugging me so much that now I'm pretty much unable to keep watching the tutorial without thinking about it. 
I understand ∂x means change in x and I picture it as the length of the horizontal side of an infinitesimally small triangle drawn over the line of the function at the point x, y (you know, the typical triangle they draw when they first teach you calculus). 
So I get why ∂y/∂x is written as a fraction and I seem to recall some cases in which operations are performed in it as if it were a fraction, which I'm fine with.
What really confuses me is that now I see this fraction written in the tutorial:

So he moved the x part of the derivative outside the fraction and even added a multiplying symbol (which is what I understand by the blue dot) as if ∂ was multiplying the expression on the right.
So my question is, can ∂x or ∂y be considered multiplications in any situation just like ∂y/∂x can be a fraction in some situations? Or is this just plain convention?
 A: No, $\partial x$ cannot be understood as a product. If it could be, then you would get
$$\frac{\partial y}{\partial x} = \frac{y}{x}$$
which obviously is not true in general.
The expression $\frac{\partial}{\partial x}$ is also known as derivation operator. This can be understood as follows:
Given a function of several variables, say $f(x,y)$, partial derivation with respect to $x$ can be seen as a map that maps the function $f$ to the function $\partial f/\partial x$. If you do so, you notice that this is a linear map, since the functions form a vector space, and
$$\frac{\partial(\alpha f + \beta g)}{\partial x} = \alpha \frac{\partial f}{\partial x} + \beta \frac{\partial g}{\partial x}$$
Such linear functions on vector spaces are also known as operators. Now traditionally, the application of an operator to a vector is written like a product; this is because the prototype of this operation is multiplying a matrix with a vector.
So if you want to write down the derivation operator, you need a notation for it. And one common notation is to write the derivation operator as $\partial/\partial x$. The reason is probably exactly because it looks like application of a normal multiplication rule, so that
$$\frac{\partial}{\partial x}f = \frac{\partial f}{\partial x}$$
and thus the rule is easy to remember and apply. On the other hand, as your question shows, it can easily mislead.
Another common notation for the partial derivative operator is $\partial_x$. This notation is usually preferred in the context of differential geometry. It has the advantage that you are not as easily misled about its meaning (and that is is less to write/type; for reasons that are outside the scope of this answer it's also a very natural notation in differential geometry), but has the slight disadvantage that you're less likely to figure out the meaning of $\partial_x f$ than of $(\partial/\partial x)f$ if you are not familiar with it.
