Why aren't placeholders for arguments more common?

Question

When learning about differentiation and integration, one often deals with functions, and it's common to use $D(x^2) = 2x$ as a function instead of $D(x\mapsto x^2) = (x\mapsto 2x)$, while it would make more sense to use placeholders such as $D(\cdot^2)(x) = (2\cdot)(x) = (2x)$ to mark the difference. While it's useful to use $x$ as a placeholder, this is often confusing when learning about functions with functions as their argument. So why isn't it more common to use such a placeholder when describing functions?