Let f : [ 0 , 1 ] → R f:[0,1] \to R possess a finite approximate derivative f ap ′ f_{\operatorname {ap}}’ Let E E be the set of points x x where f f is actually differentiable. It is shown that for every λ \lambda if { x : f ap ′ ( x ) = λ } ≠ ∅ \{ x:f_{\operatorname {ap}}’(x) = \lambda \} \ne \emptyset , then { x : f ap ′ ( x ) = λ } ∩ E ≠ ∅ \{ x:f_{\operatorname {ap}}’(x) = \lambda \} \cap E \ne \emptyset . A strengthening of the mean value theorem associated with approximate derivatives is an immediate corollary.