The first cohomology group of the trivial extension of a monomial algebra

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 $H{H}_1(A)$ and $\text{Alt}(DA)$ of the first Hochschild cohomology group $H{H}^1(TA)$. From these a formula for the dimension of $H{H}^1(TA)$ can be derived.