We estimate the derivative of a probability density function defined on [0,∞). For
this purpose, we choose the class of kernel estimators with asymmetric gamma kernel
functions. The use of gamma kernels is fruitful due to the fact that they are nonnegative, change their shape depending on the position on the semi-axis and possess
good boundary properties for a wide class of densities. We find an optimal bandwidth
of the kernel as a minimum of the mean integrated squared error by dependent data
with strong mixing. This bandwidth differs from that proposed for the gamma kernel
density estimation. To this end, we derive the covariance of derivatives of the density
and deduce its upper bound. Finally, the obtained results are applied to the case of a
first-order autoregressive process with strong mixing. The accuracy of the estimates
is checked by a simulation study. The comparison of the proposed estimates based on
independent and dependent data is provided.