Dear Syed,
you can find the answer e.g. from Section 2.7.3 in the book "Fractional Differential Equations" by I. Podlubny. The formula is quite complicated, since in general you need to calculate derivatives of all (non-negative integer) orders for the functions f and g in h(x)=f(g(x)). This is so, because the binomial coefficients in the Leibniz rule do not vanish for non-integer order derivatives.
Dear Jukka
Thanks for your reply. Can we also apply the local fractional derivatives like D^v f(g(x))=f(g(x))^(1-v) D^v g(x) ?
BR
Dear Praveen Agarwal
Could you please share the answer here or email me at [email protected]
Indeed, I will be very much thankful
.
"fractional derivative" (FD) does not mean much technically as long as the exact formalism for FD is not specified ...
for the fractional derivative of a composite function (under one specified formalism), see
http://www.rowan.edu/open/depts/math/osler/Fractional_Der_of_Composite_Function.pdf
(warning : all the pages are there ... but in the _wrong_ order !)
to tackle the difficulties in defining a satisfying FD formalism (are we happy with a non-zero FD of a constant ? this will depend of our goals when introducing the FD !), i found the papers below very useful :
http://www.sciencedirect.com/science/article/pii/S0893965908001638
http://www.worldscientific.com/doi/pdfplus/10.1142/S0217979213300053
.
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Dear Fabrice,
I thank you for emphasizing the definition of the fractional order derivative. Indeed, there are several definitions as we all know. My first answer concerned perhaps the most well-known definition in the sense of Riemann-Liouville. Similar formula holds also for the Caputo derivative (which kills constants) and that can be found e.g. from the book "The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type" by Kai Diethelm.
Both of these formulae are quite complicated and involve infinitely many terms in general. The same seems to be true for the formula given in the article of T. J. Osler. But all of these are quite natural generalizations of the chain rule.
I thank you once again for informing me on those articles given in your answer.
The derivative of the order $N$ of the function $f(z)=F(h(z))$
$$D^{N}_zf(z)=\sum_{n=0}^N\bigg\{\frac{U_n(z)D^{m}_{h(z)}f(z)}{n!}\bigg\}$$
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