  There is an increasing interest in designing differentiators, which converge
exactly before a prespecified time regardless of the initial conditions, i.e.,
which are fixed-time convergent with a predefined Upper Bound of their Settling
Time (UBST), due to their ability to solve estimation and control problems with
time constraints. However, for the class of signals with a known bound of their
$(n+1)$-th time derivative, the existing design methodologies yield a very
conservative UBST, or result in gains that tend to infinity at the convergence
time. Here, we introduce a new methodology based on time-varying gains to
design arbitrary-order exact differentiators with a predefined UBST. This UBST
is a priori set as one parameter of the algorithm. Our approach guarantees that
the UBST can be set arbitrarily tight, and we also provide sufficient
conditions to obtain exact convergence while maintaining bounded time-varying
gains. Additionally, we provide necessary and sufficient conditions such that
our approach yields error dynamics with a uniformly Lyapunov stable
equilibrium. Our results show how time-varying gains offer a general and
flexible methodology to design algorithms with a predefined UBST.
