Given a cluster-tilted algebra B, we study its first Hochschild cohomology
group HH^1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra
such that B is the relation extension of C, then we show that if C is
constrained, or else if B is tame, then HH^1(B) is isomorphic, as a k-vector
space, to the direct sum of HH^1(C) with k^{n_{B,C}}, where n_{B,C} is an
invariant linking the bound quivers of B and C. In the representation-finite
case, HH^1(B) can be read off simply by looking at the quiver of B.