Description of single-valued extensions of derivations in algebra measurable functions on interval $[a,b]$

Let $\mathbb K$ be the field of complex (real) numbers $\mathbb C$ (respectively, $\mathbb R$) and let $S_{\mathbb K}[a,b]$ be the algebra of all (classes of) measurable functions on a interval $[a,b]$. We show that the algebra of all classes $f \in S_{\mathbb K}[a,b]$, for which almost everywhere exists a finite approximative derivation $f_{ap}'$ is the maximal algebra, up to which the classical derivation $\frac {df}{dt}$ extends uniquely and any larger subalgebra of $S_{\mathbb K}[a,b]$ admits non-unique extension of this derivation, in particular, there are infinitely many extension of the classical derivation up to a derivation, defined on the whole subalgebra $S_{\mathbb K}[a,b]$.