Evaluate coefficients of delta functions in a given expression

Question

Suppose that I have a function of the form:

myfn[x_] = smoothfn[x] + A[x] DiracDelta[x-x0] + B[x] DiracDelta'[x-x0],

where smoothfn is some smooth function.

It is a well-known result that for any smooth function $F(x)$, we have the following properties:

$$ F(x) \delta(x-x_0) = F(x_0) \delta(x-x_0), \qquad \qquad \qquad \qquad F(x) \delta'(x-x_0) = -F'(x_0) \delta(x-x_0) + F(x_0) \delta'(x-x_0). $$

(This can be checked by integrating both sides against an arbitrary test function). I would like to implement a function in Mathematica, myop, that takes a function as its first argument (presumably one would need to reserve a second argument to tell it what the variable is) and returns the function so that all the coefficients in front of the Dirac functions (and their derivatives) are evaluated at $x_0$. In the above example, I would want something of the type:

myop[myfn[x],x]
(* Out: smoothfn[x] + (A[x0]-B'[x0]) DiracDelta[x-x0] + B[x0] DiracDelta'[x-x0] *)

Is there a simple way to achieve this?