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.