  We give an affirmative answer to a question of K. Ciesielski by showing that
the composition $f\circ g$ of two derivatives $f,g:[0,1]\to[0,1]$ always has a
fixed point. Using Maximoff's Theorem we obtain that the composition of two
$[0,1]\to[0,1]$ Darboux Baire-1 functions must also have a fixed point.
