  This paper focuses on effectivity aspects of the L\"uroth's theorem in
differential fields. Let $\mathcal{F}$ be an ordinary differential field of
characteristic 0 and $\mathcal{F}<u>$ be the field of differential rational
functions generated by a single indeterminate $u$. Let be given non constant
rational functions $v_1,...,v_n\in \mathcal{F}<u>$ generating a differential
subfield $\mathcal{G}\subseteq \mathcal{F}<e u>$. The differential L\"uroth's
theorem proved by Ritt in 1932 states that there exists $v\in \mathcal G$ such
that $\mathcal{G}= \mathcal{F}<v>$. Here we prove that the total order and
degree of a generator $v$ are bounded by $\min_j \textrm{ord} (v_j)$ and
$(nd(e+1)+1)^{2e+1}$, respectively, where $e:=\max_j \textrm{ord} (v_j)$ and
$d:=\max_j \textrm{deg} (v_j)$. As a byproduct, our techniques enable us to
compute a L\"uroth generator by dealing with a polynomial ideal in a polynomial
ring in finitely many variables.
