Derivative of a Logarithm

 

DERIVATION

 

We start with the function for the logarithm and calculate a change in “y” caused by a change in “x” as

 

 

 

 

where we have defined a new variable

 

 

The derivative operation for the logarithm function is

 

 

where we have defined the new variable

 

 

Recalling the binominal theorem, we can write

 

 +

 

 

where the “binominal coefficients” are the combinatorials

 

 

noting the last expression above has “j” terms in the numerator.  But in any event using this expression we can write

 

 

And the limit can be expressed as

 

 

The derivative of the logarithm is thus

 

 

or for the special case of the natural logarithm

 

 

 

REVIEW OF LOGARITHMIC FORMULAE

 

A logarithm is simply an exponent.   If we begin with the following expression

 

 

then we can define the logarithm as follows

 

 

In simple parlance, we say that the logarithm is that exponent which when applied to a base, yields a particular value.

 

A trivial result from the definition of a logarithm gives us the following

 

 

or simply

 

 

The consequences of this simple definition are several, as follows