  A structure of a left-symmetric algebra on the set of all derivations of a
free algebra is introduced such that its commutator algebra becomes the usual
Lie algebra of derivations. Left and right nilpotent elements of left-symmetric
algebras of derivations are studied. Simple left-symmetric algebras of
derivations and Novikov algebras of derivations are described. It is also
proved that the positive part of the left-symmetric algebra of derivations of a
free nonassociative symmetric $m$-ary algebra in one free variable is generated
by one derivation and some right nilpotent derivations are described.
