I am trying to expand the experssion below in powers of the perturbative parameter $\epsilon$: $$ \left[ \left( \frac{\omega}{k} - V(i \epsilon \partial_k) \right)^2 - f^2(k) \right] \Psi(k) $$ with the ansatz $$ \Psi(k) = \exp\left[ \epsilon^{-1} (\psi_0(k) + \epsilon \psi_1(k) + \cdots) \right] \, .$$ By Taylor expanding the function $V$, the lowest order (unperturbed) term is $$ \left[ \left( \frac{\omega}{k} - V(i \partial_k \psi_0) \right)^2 - f^2(k)\right]\Psi(k) \, ,$$ which I have no problem with.
However, I am having difficulty reproducing the leading order term $\mathcal{O}(\epsilon)$, which is supposed to be $$ \epsilon \left\{ \partial_k \psi_1 + \frac{1}{2} \, \partial_k \ln\left[ V'(i \partial_k \psi_0) f(k) \right] \right\} \Psi(k) \, ,$$ where prime denotes derivative with respect to the argument of $V$. I couldn't seem to get anywhere close to this expression. In particular, I don't see how $f(k)$ comes into play here. Hope someone here can help me out. Thanks a lot in advance!