We give some elementary properties of generalized derivations 
defined by Bre\v{s}ar [B] and determine functorial relations
between 
$g\text{Der}_k(S, M)$, the set of generalized derivations and  
$\text{Der}_k(S, M)$, the set of derivations from a $k$-algebra $S$ to an  
$S/k$-bimodule $M$. Using this result, we give the universal mapping property 
of generalized derivations.