  Khovanov homology extends to singular links via a categorified analogue of
Vassiliev skein relation. In view of Vassiliev theory, the extended Khovanov
homology can be seen as Vassiliev derivatives of Khovanov homology. In this
paper, we develop a new method to compute the first derivative. Namely, we
introduce a complex, called a crux complex, and prove that the Khovanov
homologies of singular links with unique double points are homotopic to
cofibers of endomorphisms on crux complexes. Since crux complexes are actually
small for some links, the result enables a direct computation of the first
derivative of Khovanov homology. Furthermore, it together with a categorified
Vassiliev skein relation provides a brand-new method for the computation of
Khovanov homology. In fact, we apply the result to determine the Khovanov
complexes of all twist knots in a universal way.
