Confusion about derivative notation I have always been confused about Leibniz notation. Not the notation itself, but the fact that it treats the differential operators ($d$, $\partial$) as being multipliable. The most famous example would probably be the Schrödinger equation, which if often denoted something like this:
$$E\psi=\left(-\frac{h^2}{2m}\frac{\partial^2}{\partial x^2}+V(x)\right)\psi=-\frac{h^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi$$
The problem with this is that is makes use of multiplication to expand the expression, $(a+b)c=a*c+b*c$, which would imply that $\partial *f=\partial f$.
Personally, I would define the differential operators as functions:
$$d(f)=\lim_{h\rightarrow \infty}{\frac{f(x+h)-f(x)}{h}}$$
So the above statement would make no sense at all. First I just accepted it as mathematical laziness, but then I stumbled upon this monstrosity:

Observe that
  $$\left(v^2\frac{\partial^2}{\partial x^2}-\frac{\partial^2}{\partial t^2}\right)y=0$$
  can be factored as (which is what you probably mean by "squaring" in the question)
  $$\left(v\frac{\partial}{\partial x}+\frac{\partial}{\partial t}\right)\left(v\frac{\partial}{\partial x}-\frac{\partial}{\partial t}\right)y=0$$

What is happening here???

What is happening is that the physicists writing out the equation are using an operator notation, where $\frac{\partial}{\partial x}$ is a shorthand for the operator of taking the partial derivative in the $x$ direction of whatever appears on the right of the operator.
The reason this appears like multiplication is that $\frac{\partial}{\partial x}$ is a linear operator, so that for any objects $\mathcal{O}$ and $\mathcal{P}$ and scalar (real or complex number) $k$
$$
\frac{\partial}{\partial x} (\mathcal{O}+ \mathcal{P}) 
=\frac{\partial}{\partial x} (\mathcal{O}) + \frac{\partial}{\partial x} (\mathcal{P})\\
\frac{\partial}{\partial x} (k\mathcal{O}) 
= k\frac{\partial}{\partial x} (k\mathcal{O}) \\
$$
That first property allows you to write things that look like you are using the distributive law of multiplication over addition.
