quantum mechanics - How does one properly define the derivative of one operator-valued function?

In Quantum Mechanics we usually consider operator-valued functions: these are functions that take in real numbers and gives back operators on the Hilbert space of the quantum system.

There are several examples of these. One of them is when we work with the Heisenberg picture, where we need to consider functions $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ such that $\alpha(t)$ is the operator at time $t$.

Another example is when we deal with exponentiation of operators, like when building the time evolution operator:

$$U(t,t_0)=\exp\left(-i\frac{H}{\hbar}(t-t_0)\right),$$

Here, the $\exp$ is usually understood as being defined via the eigenvalues of $H$.

The point is, the idea of a function $\alpha : \mathbb{R}\to \mathcal{L}(\mathcal{H})$ appears quite often in Quantum Mechanics, and sometimes one needs to differentiate these. In practice we do it formally, using all the properties we would expect, but I'm curious about how one would properly define this.

If we were dealing with bounded operators, then we could use the operator norm, which is available for this kind of operator, and define the derivative as we can usually we do when there's some norm around.

The point is that in Quantum Mechanics most of the time operators are unbounded.

So in the general case, how can one define the derivative of one operator-valued function?