- Thread starter
- #1

- Mar 22, 2013

- 573

A recent post of chisigma rings me the bell of an old problem I thought of posting in a forum (either here or MMF).

Is there any particular approach to computing a closed form for derivatives of certain smooth and continuous functions of $\mathbb{R}$?

For example, it is easy to find the $n$-th derivative of $e^x$, which is $e^x$ in turn. But this gets considerably hard for some Weird functions. For example, I recall of having to fight with the derivatives of some loggamma function in order to get an asymptotic approximation for the inverse gamma function.

Leibniz's rule in general reduces considerable amount of work, although rational function, even after Leibniz rule applied can give one a hard time, especially the ones with denominators made of function composition. Speaking of the devil, is there any general $n$-derivative closed form for composition of functions?

Nevertheless, I think an approach would use a considerable amount of tools from fractional calculus. I give my apologies for not being able to show research effort on this question, I reckon I am getting lazy lately. I hope I would be able to work out something tomorrow. Meanwhile, please feel free to post any ideas / works you are familiar with. I would be happy to see any (well-accepted) literature on this.

Is there any particular approach to computing a closed form for derivatives of certain smooth and continuous functions of $\mathbb{R}$?

For example, it is easy to find the $n$-th derivative of $e^x$, which is $e^x$ in turn. But this gets considerably hard for some Weird functions. For example, I recall of having to fight with the derivatives of some loggamma function in order to get an asymptotic approximation for the inverse gamma function.

Leibniz's rule in general reduces considerable amount of work, although rational function, even after Leibniz rule applied can give one a hard time, especially the ones with denominators made of function composition. Speaking of the devil, is there any general $n$-derivative closed form for composition of functions?

Nevertheless, I think an approach would use a considerable amount of tools from fractional calculus. I give my apologies for not being able to show research effort on this question, I reckon I am getting lazy lately. I hope I would be able to work out something tomorrow. Meanwhile, please feel free to post any ideas / works you are familiar with. I would be happy to see any (well-accepted) literature on this.

Last edited: