I was wondering if it would make sense to define a (generic) total differential operator as follows:
$$ \frac{d}{d\alpha} = \frac{\partial}{\partial\alpha} + A \, d\alpha \tag{1}\label{1} $$
where $\alpha$ is the differentiation parameter and $A$ is a generic term, not explicitly dependent on $\alpha$.
The problem with this is that I'm not sure whether it is mathematically correct to have a differential operator that is linearly dependent on the differential of the derivation parameter itself, i.e. $d\alpha$. Or rather, that includes a term directly proportional to $d\alpha$.
I know of the theta operator defined as:
$$ \theta = z\frac{d}{dz} $$
but I think this is different and probably not comparable to the case reported in equation $\eqref{1}$.
Thanks for any help.