  Let $L$ be a locally compact Hausdorff space. Suppose $A$ is a C$^*$-algebra
with the property that every weak-2-local derivation on $A$ is a
{\rm(}linear{\rm)} derivation. We prove that every weak-2-local derivation on
$C_0(L,A)$ is a {\rm(}linear{\rm)} derivation. Among the consequences we
establish that if $B$ is an atomic von Neumann algebra or on a compact
C$^*$-algebra, then every weak-2-local derivation on $C_0(L,B)$ is a linear
derivation. We further show that, for a general von Neumann algebra $M$, every
2-local derivation on $C_0(L,M)$ is a linear derivation. We also prove several
results representing derivations on $C_0(L,B(H))$ and on $C_0(L,K(H))$ as inner
derivations determined by multipliers.
