The paper deals with completely monotonic functions
and possessing derivatives
. Conditions are given when arithmetic operations, compositions and power series of functions and integral transforms with general kernel yield the completely monotonic functions. The results obtained are applied to establish the complete monotonicity for the confluent and Gauss hypergeometric functions, for functions of Bessel and Mittag-Leffler type, and for the one-dimensional Laplace, Stieltjes, Lambert and Meijer integral transforms.