The optimal bandwidth for derivative estimation will be different from the bandwidth for density estimation. In general, every feature of a density has its own optimal bandwidth selector.
If your objective is to minimize mean integrated squared error (which is the usual criterion) there is nothing subjective about it. It is a matter of deriving the value that minimizes the criterion. The equations are given in Section 2.10 of Hansen (2009).
The tricky part is that the optimal bandwidth is a function of the density itself, so this solution is not directly useful. There are a number of methods around to try to deal with that problem. These usually approximate some functionals of the density using normal approximations. (Note, there is no assumption that the density itself is normal. The assumption is that some functionals of the density can be obtained assuming normality.)
Where the approximations are imposed determines how good the bandwidth selector is. The crudest approach is called the "normal reference rule" which imposes the approximation at a high level. The end of Section 2.10 in Hansen (2009) gives the formula using this approach. This approach is implemented in the
hns() function from the
ks package on CRAN. That's probably the best you will get if you don't want to write your own code. So you can estimate the derivative of a density as follows (using
h <- hns(x,deriv.order=1)
den <- kdde(x, h=h, deriv.order=1)
A better approach, usually known as a "direct plug in" selector, imposes the approximation at a lower level. For straight density estimation, this is the Sheather-Jones method, implemented in R using
density(x,bw="SJ"). However, I don't think there is a similar facility available in any R package for derivative estimation.
Rather than use straight kernel estimation, you may be better off with a local polynomial estimator. This can be done using the
locpoly() function from the
ks package in R. Again, there is no optimal bandwidth selection implemented, but the bias will be smaller than for kernel estimators. e.g.,
den2 <- locpoly(x, bandwidth=?, drv=1) # Need to guess a sensible bandwidth