Mean value theorem of differential calculus/From mean value theorem of integral calculus/Exercise

Prove the mean value theorem for differential calculus for differentiable functions

$g\colon \mathbb {R} \longrightarrow \mathbb {R}$

and a compact interval ${}[a,b]\subset \mathbb {R}$, using the mean value theorem of integral calculus

(you do not have to show that the average velocity is obtained in the interior of the interval).

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