The FunctionAdvisor(differentiation_rule, math_function) command returns both the differentiation rule (first derivative) and the symbolic differentiation rule ($n^{\mathrm{th}}$ derivative) for the function. The result thus consists of a sequence of two equations, with inert derivatives on the left-hand sides (represented using Diff) and the corresponding values of these derivatives on the right-hand sides.

FunctionAdvisor( differentiation_rule, arcsin );

FunctionAdvisor( differentiation_rule, dilog );

$\frac{\ⅆ}{\ⅆz}\mathrm{dilog}\left(z\right)=\frac{\ln \left(z\right)}{1-z},\frac{{\ⅆ}^n}{\ⅆz^n}\mathrm{dilog}\left(z\right)=\left\{\begin{array}{cc}\mathrm{dilog}\left(z\right)& n=0\\ {\left(\mathrm{-1}\right)}^n\mathrm{MeijerG}\left(\left[\left[1-n\,1-n\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[-n\right]\right]\,z-1\right)& \mathrm{otherwise}\end{array}\right.$

eq1 := FunctionAdvisor(diff, Zeta);

* Partial match of "diff" against topic "differentiation_rule".

$\mathrm{eq1}≔\frac{\ⅆ}{\ⅆs}{\mathrm{\zeta }}^{\left(n\right)}\left(s\,a\right)={\mathrm{\zeta }}^{\left(n+1\right)}\left(s\,a\right),\frac{\ⅆ}{\ⅆa}{\mathrm{\zeta }}^{\left(n\right)}\left(s\,a\right)=-s{\mathrm{\zeta }}^{\left(n\right)}\left(s+1\,a\right)-\left(\left\{\begin{array}{cc}0& n=0\\ n{\mathrm{\zeta }}^{\left(n-1\right)}\left(s+1\,a\right)& \mathrm{otherwise}\end{array}\right.\right)$

has([eq1],a), has([eq1],b), has([eq1],z);

$\mathrm{false},\mathrm{false},\mathrm{false}$

eq2 := FunctionAdvisor( diff, Zeta(a,b,z) );

* Partial match of "diff" against topic "differentiation_rule".

$\mathrm{eq2}≔\frac{\ⅆ}{\ⅆb}{\mathrm{\zeta }}^{\left(a\right)}\left(b\,z\right)={\mathrm{\zeta }}^{\left(a+1\right)}\left(b\,z\right),\frac{\ⅆ}{\ⅆz}{\mathrm{\zeta }}^{\left(a\right)}\left(b\,z\right)=-b{\mathrm{\zeta }}^{\left(a\right)}\left(b+1\,z\right)-\left(\left\{\begin{array}{cc}0& a=0\\ a{\mathrm{\zeta }}^{\left(a-1\right)}\left(b+1\,z\right)& \mathrm{otherwise}\end{array}\right.\right)$

has([eq2], a), has([eq2], b), has([eq2], z);

$\mathrm{true},\mathrm{true},\mathrm{true}$

FunctionAdvisor( diff, Zeta(z) );

* Partial match of "diff" against topic "differentiation_rule".

$\frac{\ⅆ}{\ⅆz}\mathrm{\zeta }\left(z\right)={\mathrm{\zeta }}^{\left(1\right)}\left(z\right),\frac{{\ⅆ}^n}{\ⅆz^n}\mathrm{\zeta }\left(z\right)={\mathrm{\zeta }}^{\left(n\right)}\left(z\right)$

FunctionAdvisor( differentiation_rule, Ei(z) );

$\frac{\ⅆ}{\ⅆz}\mathrm{Ei}\left(z\right)=\frac{{\ⅇ}^z}{z},\frac{{\ⅆ}^n}{\ⅆz^n}\mathrm{Ei}\left(z\right)=\left\{\begin{array}{cc}\mathrm{Ei}\left(z\right)& n=0\\ -{\ⅇ}^z\left(\sum _{\mathrm{\_k1}=0}^{n-1}\frac{\left(n-1\right)!{\left(\mathrm{-1}\right)}^{n-\mathrm{\_k1}}{z}^{\mathrm{\_k1}-n}}{\mathrm{\_k1}!}\right)& \mathrm{otherwise}\end{array}\right.$

FunctionAdvisor( differentiation_rule, Ei(a,z) );

$\frac{\ⅆ}{\ⅆa}{\mathrm{Ei}}_a\left(z\right)=-z^a\mathrm{MeijerG}\left(\left[\left[\right]\,\left[0\,0\right]\right]\,\left[\left[\mathrm{-1}\,\mathrm{-1}\,-a\right]\,\left[\right]\right]\,z\right),\frac{\ⅆ}{\ⅆz}{\mathrm{Ei}}_a\left(z\right)=-{\mathrm{Ei}}_{a-1}\left(z\right),\frac{{\ⅆ}^n}{\ⅆz^n}{\mathrm{Ei}}_a\left(z\right)=-\frac{{\left(\mathrm{-1}\right)}^n\mathrm{MeijerG}\left(\left[\left[0\,a\right]\,\left[\right]\right]\,\left[\left[0\right]\,\left[a-1\,n\right]\right]\,z\right)}{{\left(-z\right)}^n}+\frac{\mathrm{\pi }{z}^{a-1-n}}{\mathrm{\Gamma }\left(a-n\right)\sin \left(\mathrm{\pi }a\right)}$

The FunctionAdvisor/differentiation_rule command was updated in Maple 2016.