# On the Leibniz rule and Laplace transform for fractional derivatives

## TL;DR

This paper uses Taylor series to analyze fractional derivatives, revealing limitations of Leibniz rule and Laplace transform applications for Caputo and Riemann-Liouville derivatives, respectively.

## Contribution

It provides exact formulas for Caputo Leibniz rule and clarifies the Laplace transform of Riemann-Liouville derivatives using series representation.

## Key findings

- Leibniz rule does not apply to Caputo derivatives
- Laplace transform of Riemann-Liouville derivatives is valid only for sufficiently smooth functions
- Series representation confirms the derived formulas through examples

## Abstract

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results.