If $G$ is a $k-$ group scheme (seeing as a functor) exist a good definition of what is the derived group scheme? (Or a good reference for a good definition). Where derived I'm talking in the sense of group theory.


PS:We may assume that $G$ is of finite type and smooth.



Derived Group Scheme
The following quote, from Pseudo-reductive groups by Conrad, Gabber and Prasad (Definition A.1.14, lightly edited) should answer your question.



> 
> The derived group D(G) of a smooth group G of finite type
>  over a field k is the unique smooth closed k-subgroup such that (D(G))(K) is
>  the commutator subgroup of G(K) for any algebraically closed extension K/k.
> 
> 
> Note that the derived group exists without connectedness hypotheses on
>  G; see [Bo2, I, 2.4] for the affine case and [SGA3,
>  VI B , 7.2(vii), 7.10] for the general case. The formation of D(G) commutes
>  with any extension of the base field, and the quotient map G → G/D(G) is
>  initial among all k-homomorphisms from G to a commutative k-group scheme
>  (see Lemma 5.3.4 for a generalization).
> 
> 
> 


[Bo2] is the same reference to Borel's "Linear Algebraic Groups" that was provided in a comment by Uri Bader.


