Differential operator - what is $\frac{\partial f}{\partial x}x$? given  $f$ - a smooth function, $f\colon\mathbb{R}^2\to \mathbb{R}$.
I have a differential operator that takes $f$ to $\frac{\partial f}{\partial x}x$, but I am unsure what this is.
If, for example, the operator tooked $f$ to $\frac{\partial f}{\partial x}$ then I understand that this operator derives by $x$, $\frac{\partial f}{\partial x}x$ derives by $x$ and then what ?
Can someone please help with this ? (I'm confused...)

Let $D$ be a (first order) differential operator (i.e. a derivation) and let $h$ be a function. (In your example $h$ is the function $x$ and $D$ is $\frac{\partial}{\partial x}$). I claim that there is a differential operator $hD$ given by
$$
f \mapsto hDf.
$$
(This means multiply $h$ by $Df$ pointwise.) For this claim to hold, we just need it to be true that the Leibniz (product) rule holds, i.e.
$$
hD(fg) = f(hD)g + g(hD)f.
$$
Since $D$ itself satisfies the Leibniz rule, this is true. (We also need the operator $hD$ to take sums to sums, which it does.)
Put less formally, to compute $x \frac{\partial}{\partial x}$ on a function $f$, take the partial with respect to $x$, then multiply what you get by $x$.
