  In this paper, we investigate the existence and characterizations of the
Fr\'echet derivatives of the solution to time-harmonic elastic scattering
problems with respect to the boundary of the obstacle. Our analysis is based on
a technique - the factorization of the difference of the far-field pattern for
two different scatterers - introduced by Kress and Pa\"ivarinta to establish
Fr\'echet differentiability in acoustic scattering. For the Dirichlet boundary
condition an alternative proof of a differentiability result due to
Charalambopoulos is provided and new results are proven for the Neumann and
impedance exterior boundary value problems.
