Here I summarise the recursive rules for deriving (differentiating) compound expressions.
I will also write
i.e. composition of functions with a circle
i.e. multiplicative product of functions with a dot or juxtaposition;
these are both products but in different senses.
Note that mathematical notation can be a bit ambiguous or inconsistent. For example:
is the inverse function (compositionally), not the reciprocal (which is written ); but
is the (multiplicative) square of the result of the function.
This is a nearly minimal set of basic rules. Such rules are usually established from limit arguments on the definition of derivation.
[const] , (i.e. for constant with respect to x);
From these a little algebra may be used to obtain:
(from [const,+]: );
(from [const,×]: );
(from [const,o]: ).
These in turn lead to:
Yet More Rules
These all lead to, for example: