The derivative of an exponential operator is defined:
$$\frac{\partial}{\partial n}\exp[- iH] = -i\int_0^1d\alpha \;\exp[-i(1-\alpha)H]\frac{\partial H}{\partial n} \exp[-i\alpha H].$$
What then is the following:
$$\frac{\partial}{\partial m}\frac{\partial}{\partial n}\exp[- iH],$$
where $H$ is a function of $m$ and $n$. This is a study of identities, and finds applications in many Physically motivated situations. My motive is to find the nth derivative of the identity given in the first line.