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I was reading through the online help of WolframAlpha (link) and found this statement:

Wolfram|Alpha calls Mathematica's $D$ function, which uses a table of identities much larger than one would find in a standard calculus textbook. It uses ”well known” rules such as the linearity of the derivative, product rule, power rule, chain rule, so on. Additionally, $D$ uses ”lesser known” rules to calculate the derivative of a wide array of special functions.

What could these "lesser known" rules be?

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    $\begingroup$ Maybe a question for Mathematica SE $\endgroup$ – Simon S Apr 29 '15 at 23:03
  • $\begingroup$ And what are the "special functions"? E.g. I use $(\ln f)'=\frac{f'}{f}$ in the form of $f'=f\cdot(\ln f)'$ for things like, say, $\frac{(x-1)(x+2)(x-3)}{(x-2)^2}$. $\endgroup$ – Alexey Burdin Apr 29 '15 at 23:12
  • $\begingroup$ Perhaps one such "lesser known rules" is General Leibnitz rule. $\endgroup$ – Mark Viola Apr 29 '15 at 23:15
  • $\begingroup$ I was indeed hoping for a few rules being more obscure than the general Leibniz rule. So far the answers suggest compiling identities of the special functions involving their derivatives. That is nice, but those seem to hold only for a few special functions and not a "wide array". Maybe it is just advertising exaggeration. :-) $\endgroup$ – mvw Apr 29 '15 at 23:49
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It's going to mean a large proportion of the identities in the DLMF, for one. Or perhaps more appropriately, the Wolfram functions site, including things like $$ \Gamma'(s) = \Gamma(s)\psi(s) $$ for the Gamma-function, Bessel function things like $$ J_n'(x) \frac{1}{2} (J_{n-1}(x)-J_{n+1}(x)), $$ orthogonal polynomials: $$ P_n^{(a,b)}(x) = \frac{1}{2} (a+b+n+1) P_{n-1}^{(a+1,b+1)}(x), $$ elliptic functions: $$ \frac{d}{dx} \operatorname{sn}{(x\mid m)} = \operatorname{cn}{(x|m)} \operatorname{dn}{(x|m)}, $$ hypergeometric functions: $$ \frac{d}{dx} {}_3F_3(a,b,c;d,e,f;x) = \frac{a b c \, {}_3F_3(a+1,b+1,c+1;d+1,e+1,f+1;x)}{d e f}, $$ and functions you've probably never heard of: $$ \text{gd}'(x) = \operatorname{sech}{x} \\ (\text{gd}^{-1})'(x) = \sec{x} \\ W'(x) = \frac{W(x)}{x (W(x)+1)} $$

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  • $\begingroup$ $W$ shows sometimes up here on SE, but gd indeed was new to me, thanks! Nice links. $\endgroup$ – mvw Apr 29 '15 at 23:40
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For example, for spherical Bessel functions $$ \frac{d}{dz}j_n(z) = j_{n-1}(z) - \frac{n+1}{z}j_n(z) $$ Many such relations can be found in Abromowitz and Stegun.

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