Given the function $ x^{a}\log(x) $ natural logarithmic

Could someone tell me how to evaluate the fractional derivative

$$ \frac{d^{b}}{dx^{b}}x^{a}\log(x) $$ for positive $a$ and $b$

  • 1
    $\begingroup$ Which fractional derivative are you considering? The main difficulty here is that the Leibniz rule doesn't hold for fractional-order derivatives. There are however certain extensions. For the Riemann-Liouville case, check out Section 2.7.2 in: I. Podlubny, "Fractional Differential Equations," Academic Press, 1999. For the Caputo derivative see Eq. (9) in K. Diethelm et al., "Algorithms for the fractional calculus: A selection of numerical methods," Comp. meth. appl. mech. eng. 194 (2005). $\endgroup$ – Pantelis Sopasakis Dec 10 '14 at 11:15

There is an extension to the fractional derivative for the General Leibniz rule:


For your case,



  • $\begingroup$ Of course, this probably isn't the only extension to the Leibniz rule, as Pantelis Sopasakis explains. $\endgroup$ – Simply Beautiful Art Jun 17 '16 at 18:44

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