In this paper we consider the problem of bandwidth selection in local polynomial estimation of derivative functions. We use a dependent data context, and analyze time series which are realizations of strictly stationary processes. We consider the estimation of the first derivative of the conditional mean function for a non-linear autoregressive model. First of all, we emphasize the role assumed by the smoothing parameter, by showing how the choice of the bandwidth is crucial for the consistency of the non-parametric estimation procedure, through an example on simulated data. We then use a new approach for the selection of such a parameter, based on the neural network technique. Such alternative method presents several advantages with respect to the traditional approach used so far.