# The Causal Fractional Derivatives

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## Abstract

The fractional calculus is a 300 years old mathematical discipline. In fact and some time after the publication of the studies on Differential Calculus, where he introduced the notation \( {\frac{{{\text{d}}^{n} y}}{{{\text{d}}x^{n} }}} ,\) Leibnitz received a letter from Bernoulli putting him a question about the meaning of a non-integer derivative order. Also he received a similar enquiry from L’Hôpital: *What if n is* ½? Leibnitz’s replay was prophetic: *It will lead to a paradox*, *a paradox from which one day useful consequences will be drawn*, *because there are no useless paradoxes*. It was the beginning of a discussion about the theme that involved other mathematicians like Euler and Fourier. Euler suggested in 1730 a generalisation of the rule used for computing the derivative of the power function. He used it to obtain derivatives of order 1/2. Nevertheless, we can say that the XVIII century was not proficuous in which concerns the development of Fractional Calculus. Only in the early XIX, interesting developments started being published. Laplace proposed an integral formulation (1812), but it was Lacroix who used for the first time the designation “derivative of arbitrary order” (1819).

## Keywords

Fractional Derivative Fractional Calculus Laplace Transform Fractional Integration Caputo Derivative## References

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