To enhance accuracy of robot state estimation, active sensing (or perception-aware) methods seek trajectories that maximize the information gathered by the sensors. To this aim, one possibility is to seek trajectories that minimize the (estimation error) covariance matrix output by an extended Kalman filter (EKF), w.r.t. its control inputs over a given horizon. However, this is computationally demanding. In this article, we derive novel backpropagation analytical formulas for the derivatives of the covariance matrices of an EKF w.r.t. all its inputs. We then leverage the obtained analytical gradients as an enabling technology to derive perception-aware optimal motion plans. Simulations validate the approach, showcasing improvements in execution time, notably over PyTorch's automatic differentiation. Experimental results on a real vehicle also support the method.