My course problem booklet (mathematics BSc, second year module in algebra, unpublished) has a question,
Let $D:=x\frac{\text{d}}{\text{d}x}$ be a linear operator on $\mathbb{R}[x]$. Calculate an expression for $D^2$ in terms of $x$ and $\frac{\text{d}}{\text{d}x}$.
The solution booklet has \begin{aligned} D^2f & = x\frac{\text{d}}{\text{d}x}\left(x\frac{\text{d}f}{\text{d}x}\right) \\ & = x\left[\frac{\text{d}f}{\text{d}x}+x\frac{\text{d}^2f}{\text{d}x^2}\right]. \end{aligned}
I can't immediately understand this in first-principle terms. It looks to me like an application of the product rule, but we haven't derived the product rule in the context of linear operators on vector spaces, so I'm not sure why it should be obvious that it can be applied. Can anyone help me understand?
Yes, it’s an application of the product rule, what’s so surprising? $\mathbb{R[x]}$ is just the set of all real polynomials and polynomials satisfy the product rule.
There’s nothing being used about linear operators, what’s being used is that $d/dx$ satisfies the product rule.
Judging from your comment, your problem seems to be that that the product rule is being applied to $x\times df/dx$ but that’s no problem because both $x$ and $df/dx$ are differentiable polynomial functions [because $f$ is a polynomial].