This paper investigates the unknown input observability problem in the nonlinear case. The analysis starts by extending the observability rank criterion. This is obtained by a state augmentation and is called the extended observability rank criterion (first contribution). In the case of a single unknown input, the paper provides an analytical method to directly obtain the entire observable codistribution (second and main contribution). As in the standard case of only known inputs, the observable codistribution is obtained by recursively computing the Lie derivatives along the vector fields that characterize the dynamics. On the other hand, in correspondence of the unknown input, the corresponding vector field must be suitably rescaled. Additionally, the Lie derivatives must be computed also along a new set of vector fields obtained by performing suitable Lie bracketing of the vector fields that define the dynamics. In practice the entire observable codistribution is obtained by a very simple recursive algorithm. However, the analytic derivations required to prove that this codistribution fully characterizes the state observability properties are complex. Finally, it is shown that the recursive algorithm converges in a finite number of steps and the criterion to establish that the convergence has been reached is provided (third contribution). Also this proof is based on several tricky analytical steps. The proposed approach is used to derive the observability properties of several systems, starting from simple ones, i.e., characterized by a single unknown input. The last application is a very complex unknown input observability problem and is characterized by four unknown inputs, two known inputs and at least two outputs. Specifically, we derive the observability properties for the visual-inertial structure from motion problem in the case when part of the inertial inputs are missing.