In this note, we discuss, improve and complement some recent results of the conformable fractional derivative introduced and established by Katugampola[arxiv:1410.6535v1] and Khalil et al. [J. Comput. Appl. Math. 264(2014) 65-70]. Among other things we show that each function $f$ defined on $(a,b)$, $a>0$ has a conformable fractional derivative (CFD) if and only if it has a classical first derivative. At the end of the paper, we prove the Rolle's, Cauchy, Lagrange's...