The usefulness of sensitivity analyses in mechanical engineering is very well-known. Interesting examples of sensitivity analysis applications include the computation of gradients in gradient-based optimization methods and the determination of the parameter relevance on a specific response or objective. In the field of multibody dynamics, analytical sensitivity methods tend to be very complex, and thus, numerical differentiation is often used instead, which degrades numerical accuracy. In this work, a simple and original method based on state-space motion differential equations is presented. The number of second-order motion differential equations equals the number of DOFs, that is, there is one differential equation per independent acceleration. The dynamic equations are then differentiated with respect to the parameters by using automatic differentiation and without manual intervention from the user. By adding the sensitivity equations to the dynamic equations, the forward dynamics and the independent sensitivities can be robustly computed using standard integrators. Efficiency and accuracy are assessed by analyzing three numerical examples (a double pendulum, a four-bar linkage, and an 18-DOF coach) and by comparing the results with those of the numerical differentiation approach. The results show that the integration of independent sensitivities using automatic differentiation is stable and accurate to machine precision.