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?