Based on my understanding when doing quantum mechanics we deal with a small set of mathematical objects: namely scalars, kets, bras, and operators. But then in the Schrodinger equation we have this time-derivative thing which looks an awful lot like an operator, but from what I've been told it's not considered an operator. My question, then, is what exactly is a time-derivative in quantum mechanics and why is it not considered an operator.
Edit: The issue that this is a possible duplicate of threads linked in the comments below has been brought up. I saw the threads linked below before I posted this question, and I thought this question was sufficiently different because I'm not interested in a time-operator per se, but rather what $\partial_t$ is mathematically. I've been told it's not an operator in the same way the momentum operator is, but it does look like an operator that scales the state vector proportional to its energy.