We consider the nonparametric estimation of the multivariate probability density function and its partial derivative with a support on $[0,\infty)$ by dependent data. We use the class of gamma-kernel estimators which are asymmetric. The gamma kernels are nonnegative and change their shape depending on the position on the semi-axis. They possess good boundary properties for a wide class of densities useful in many applications like engineering, signal processing, actuarial science etc. The theoretical asymptotic properties of the multivariate density and its partial derivative estimates like biases, variances and covariances are derived. We obtain the optimal bandwidth selection for both estimates as a minimum of the mean integrated squared error (MISE) assuming dependent data with a strong mixing. Optimal rates of convergence of the MISE are found.