We discuss the uniform computational complexity of the derivatives of C∞-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function g∈C∞[−1,1] such that the sequence is not bounded by any recursive function. On the other hand, we show that if f∈C∞[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f(n)(x)| is bounded by 2p(n) for all x∈[−1,1] for some polynomial p, then the sequence is uniformly polynomial time computable.