On the basis of quasiderivatives and the upper triangular Hessenberg form, an approach to investigating the observability problem for linear time-varying systems is proposed. The quasiderivatives are defined for some lower triangular matrix, and the simplest rules of the quasidifferentiation are described. Conditions for linear independence of continuous quasidifferentiable functions are established in terms of the Wronski matrix. The necessary and sufficient conditions for an upper triangular Hessenberg form in the orbit of linear time-varying systems are obtained, and the method to find such systems is described. New efficient conditions for various types of observability (complete, differential, uniform) are obtained.