  In this work we give an explicit solution to the problem of differentiation
of hyperelliptic functions in genus $4$ case. It is a genus $4$ analogue of the
classical result of F. G. Frobenius and L. Stickelberger [F. G. Frobenius, L.
Stickelberger, "Uber die Differentiation der elliptischen Functionen nach den
Perioden und Invarianten", J. Reine Angew. Math., 92 (1882), 311-337] in the
case of elliptic functions. An explicit solution in the genus $2$ case was
given in [V. M. Buchstaber, "Polynomial dynamical systems and Korteweg-de Vries
equation", Proc. Steklov Inst. Math., 294 (2016), 176-200]. An explicit
solution in the genus $3$ case was given in [E. Yu. Bunkova, "Differentiation
of genus 3 hyperelliptic functions", European Journal of Mathematics, 4:1
(2018), 93-112].
