We study the Banach {}^* -algebra C^1_{\mathrm{op}}(I) of C^1 -functions on the compact interval I such that the corresponding Hilbert space operator function T → f(T) , for T = T^* and \mathrm{sp}(T) ⊂ I , is Fréchet differentiable. If f(x) = \int e^{itx} \hat f^(t)dt , we know that the differential is given by the formula df_T(S) = \int_{-∞}^∞ \int_0^1 U_{st}SU_{(1-s)t} ds\widehat{f'}(t) dt, where U_t = \exp(itT) . Functions of this type are dense in C^1_{\mathrm{op}}(I) , and C^2(I) ⊂ C^1_{\mathrm{op}}(I) ⊂ C^1(I) , so several classical results can be deduced. In particular we show that if T \in \mathfrak D(δ) , where δ is the generator of a one-parameter group of {}^* -automorphisms of a C^* -algebra \mathfrak A (or just a closed {}^* -derivation in \mathfrak A ), then f(T) \in \mathfrak D(δ) for every f in C^1_{\mathrm{op}}(I) , where \mathrm{sp}(T) ⊂ I , and δ(f(T)) = df_T(δ(T)).