  This paper introduces the path derivatives, in the spirit of Dupire's
functional It\^o calculus, for the controlled paths in the rough path theory
with possibly non-geometric rough paths. The theory allows us to deal with
rough integration and rough PDEs in the same manner as standard stochastic
calculus. We next study rough PDEs with coefficients depending on the rough
path itself, which corresponds to stochastic PDEs with random coefficients.
Such coefficients is less regular in the time variable and is not covered in
the existing literature. The results are useful for studying viscosity
solutions of stochastic PDEs.
