  Given a finite--dimensional monomial algebra $A$ we consider the trivial
extension $TA$ and provide formulae, depending on the characteristic of the
field, for the dimensions of the summands $HH_1(A)$ and $\Alt(DA)$ of the first
Hochschild cohomology group $HH^1(TA)$. From these a formula for the dimension
of $HH^1(TA)$ can be derived.
