Sensitivity analysis for systems governed by partial differential equations is now commonly used by engineers to assess performance modification due to parameter changes. A typical illustration concerns shape optimization procedures based on the adjoint method, used in aeronautics to improve aerodynamic or structural performance of aircrafts. However, these approaches are usually limited to first-order derivatives and steady PDE systems, due to the complexity to extend the adjoint method to higher-order derivatives and the associated reverse time integration. Alternatively, this work investigates the use of the direct differentiation approach (continuous sensitivity equation method) to estimate high-order derivatives for unsteady flows. We show how this method can be efficiently implemented in existing solvers, in the perspective of providing a Taylor expansion of the PDE solution with respect to control parameters. Applications to optimization and uncertainty estimation are finally considered.