Fractional derivative of composite functions: exact results and physical applications*

We examine the fractional derivative of composite functions and present a generalization of the product and chain rules for the Caputo fractional derivative. We derive the product rule with the expression of the Caputo fractional derivative as an infinite expansion of integer order derivatives and the chain rule as a generalization of di Bruno's formula. Unlike the Leibniz and di Bruno formulae that characterize an integer-order derivative of a product of functions and composite functions, respectively, and which result in a finite series of lower order derivatives, the fractional analogs of these formulae produce an infinite series of fractional derivatives of the constituent functions. These results are important for a comprehensive description of transport phenomena through multiscale physical systems and biological structures, e.g., porous materials, disordered media, and neuron clusters. We demonstrate the utility of these results by the evaluation of the Caputo fractional derivative of hyperbolic tangent and suggest that the application of the derived chain and product rules to elementary functions, whose Caputo fractional derivative is expressible as a generalized hypergeometric function, leads to an infinite series of generalized hypergeometric functions.