Studying quantum mechanics I have encountered several times in proofs this expression: $$ \frac{\mathrm{d}\langle\phi|\hat{H}|\psi\rangle}{\mathrm{d}t} = \biggl\langle\frac{\mathrm{d}\phi}{\mathrm{d}t}\bigg\rvert\hat{H}\bigg\lvert\psi\biggr\rangle + \biggl\langle\phi\bigg\rvert\frac{\mathrm{d}\hat{H}}{\mathrm{d}t}\bigg\lvert\psi\biggr\rangle + \biggl\langle\phi\bigg\rvert\hat{H}\bigg\lvert\frac{\mathrm{d}\psi}{\mathrm{d}t}\biggr\rangle $$
What does the derivative of an operator mean mathematically ? Is something that can be rigorously mathematically defined? Or is just the juxtaposition of the derivative and the operator?
Moreover I was wondering if it is possible to derive this form of Liebniz rule. If I explicitly do the integrals I can carry in the time derivative, but I don't understand how to apply a derivative to a mathematical object which is defined only by its action on a well defined function.
