  There is a growing interest in differentiation algorithms that converge in
fixed time with a predefined Upper Bound on the Settling Time (UBST). However,
existing differentiation algorithms are limited to signals having an $n$-th
order Lipschitz derivative. Here, we introduce a general methodology based on
time-varying gains to circumvent this limitation, allowing us to design $n$-th
order differentiators with a predefined UBST for the broader class of signals
whose $(n+1)$-th derivative is bounded by a function with bounded logarithmic
derivative. Unlike existing methods whose time-varying gain tends to infinity,
our approach yields a time-varying gain that remains bounded at convergence
time. We show how this last property maintains exact convergence using bounded
gains when considering a compact set of initial conditions and improves the
algorithm's performance to measurement noise.
