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?
On
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: $$ \text{gd}'(x) = \operatorname{sech}{x} \\ (\text{gd}^{-1})'(x) = \sec{x} \\ W'(x) = \frac{W(x)}{x (W(x)+1)} $$
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.