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We prove that if $f:I\subset \Bbb R\to \Bbb R$ is of bounded variation, thenthe noncentered maximal function $Mf$ is absolutely continuous, and itsderivative satisfies the sharp inequality $\|DMf\|_1\le |Df|(I)$. This allowsus obtain, under less regularity, versions of classical inequalities involvingderivatives.

Functions of bounded variation, the derivative of the one dimensional maximal function, and applications to inequalities