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We present a new derivation for the optimal decay of \textit{arbitrary}higher order derivatives for $L^p$ solutions to the compressible fluid model ofKorteweg type. This approach, based on Gevrey estimates, is to establishuniform bounds on the growth of the radius of analyticity of the solution innegative Besov norms. For that end, the maximal regularity property involvingGevrey multiplier of heat kernel and non standard product Besov estimates arewell developed. Our approach is partly inspired by Oliver-Titi's work and isapplicable to a wide range of dissipative systems.

Decay of higher order derivatives for $L^p$ solutions to the compressible fluid model of Korteweg type