CK12 Calculus - Single Variable

Thuswe conclude

and in the special case where

To generalize,if is a differentiablefunctionof and if then the abovetwo equations,after
the ChainRule is applied,will producethe generalizedderivativerule for logarithmicfunctions.

Derivativesof LogarithmicFunctions

Remark:Studentsoftenwonderwhy the constant is definedthe way it is. The answeris in the derivative

of With any otherbasethe derivativeof wouldbe equal
a morecomplicatedexpressionthan Thinkingback to anotherunexpectedunit, radians,the derivative
of is the simple expression only if is in radians. In degrees,

, whichis morecumbersomeand harderto remember.
Example1:

Find the derivativeof
Solution:

Since , for