How to apply this operator?

Question

Let $A$ be the operator $2(x+\partial_x)$. Suppose we have a function $f$ and that we apply the operator to this function. How this operator is applied? $2xf+\partial_xf$ or $2x+\partial_xf$? I guess is the last one but I am not sure. By the way, what kind of operator is this? Differential?

Thanks!

Answer 1

We have $A=2(x+\partial_x)$ so $Af$ = $2(x+\partial_x)f=2xf+2\partial_xf$. The trick to this sort of thing is to think of it like (non commutative) multiplication and then it almost always works.

Answer 2

The former.

$$L=2(x+\partial_x) \implies Lf = 2xf + 2\partial_x f$$

And it is indeed a differential operator.