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It is proved that any function of a Lusin-type class, the class of$ACG_r$-functions, is differentiable almost everywhere in the sense of aderivative defined in the space~$L^r$, $1\le r<\infty$. This leads to obtaininga full descriptive characterization of a Henstock-Kurzweil-type integral, the$HK_r$-integral, which serves to recover functions from their$L^r$-derivatives. The class $ACG_r$ is compared with the classical Lusin class$ACG$ and it is shown that a continuous $ACG$-function can fail to be$L^r$-differentiable almost everywhere.

On the $L^r$-differentiability of Two Lusin Classes and a Full Descriptive Characterization of the $HK_r$-integral