I have an operator defined as: $$ \exp \Bigg[ \lambda \ \frac{\partial}{\partial R} \Bigg] $$
I'm trying to optimize this with respect to the parameter $\lambda$.
In other words, I'm trying to get an expression for
$$ \frac{\partial}{\partial \lambda} \Bigg[\exp \Big[ \lambda \ \frac{\partial}{\partial R} \Big] \Bigg]$$
I read about Sneddon's formula but that doesn't seem to solve the problem. Any suggestions on how I should approach this?
$$\exp(\lambda \partial_R)f(R) = \sum_{k=0}^\infty \frac{\lambda^k f^{(k)}(R)}{k!}.$$
Taking derivatives in $\lambda$ gives:
$$\sum_{k=0}^\infty \frac{\lambda^k f^{(k+1)}(R)}{k!},$$
or $$\exp(\lambda \partial_R)f'(R).$$