How to take differential of an Operator which acts on a function? How can i take differential of an operator which acts on a function?
$d\hat A\Phi=?$

point:
$d$ (denotes differential)
$\hat A$ (denotes operator)
$\Psi$ (denotes function )
 A: Two approaches are possible, it's unclear from your post which to chose.
First option - it's just a differential of the image, i.e. $d(\hat A\Psi)$
Another approach is to adopt the usual definition of differential.
Let's suppose that your operator sends a function from one banach space to another:
$$\hat A:V_1\to V_2,\quad \Psi \in V_1,\quad \hat A\Psi \in V_2.$$ Then you can consider a linear operator $L:V_1\to V_2$
$$\hat A(\Psi+\Delta\Psi)-\hat A\Psi=L(\Delta\Psi)+o(\|\Delta\Psi\|_{V_1}) \text{ as }\Delta\Psi\to 0.$$
Then you call $L$ the differential of $\hat A$ in $\Psi$.
A: I think the most reasonable thing to assume is that $\hat A$ depends on some additional parameters and that we are considering our differential to be with repsect to these parameters.
For example a translation $\hat A_t$ which acts as $\hat A_t[\Phi](x) = \Phi(x + t)$ where $t$ is the additional parameter. Then
$$\hat A_{t + h}[\Phi] - \hat A_t[\Phi](x) = \Phi(x + t + h) - \Phi(x + t) = \Phi'(x + t)h + |h| \varepsilon(|h|)$$
or 
$$d \hat A_t[\Phi](x) = \Phi'(x + t)dt = \hat A_t D_x [\Phi](x) dt$$
which yields us something like
$$d\hat A_t = \hat A_t D_x dt$$
in this case.
