Given a diffeomorphism of the interval, we consider the uniform norm
of the derivative of its $n$-th iteration. We get a sequence of real
numbers called the growth sequence. Its asymptotic behavior is an
invariant which naturally appears both in smooth dynamics and in the
geometry of the diffeomorphism group. We find sharp estimates
for the growth sequence of a given diffeomorphism in terms of the
modulus of continuity of its derivative. These estimates extend
previous results of Polterovich--Sodin and Borichev.