find the derivative of the following function f(x)=log(x^2_1/2x)

You need to change the base of logarithm such
that:

`log_(x^2) (1/2x) = ln (1/2x)/ ln
(x^2)`

You need to differentiate the function with respect
to x, using the quotient rule:

`f(x) = ln (1/2x)/ ln (x^2)
=gt f'(x) = {[ln (1/2x)]'*ln (x^2) - ln (1/2x)*[ ln (x^2)]'}/(ln
(x^2))^2`

`f'(x) = ((((1/2x)')/(1/2x))*ln (x^2) -
(2x/x^2)*(ln (1/2x)))/((ln (x^2)^2)`

`f'(x) = ((-ln
(x^2))/(x) - (ln (1/2x)^2)/(x)) / (ln (x^2)^2)`

`f'(x) =
(ln(1/x^2) - ln (1/2x)^2)/(x*(ln (x^2))^2)`

`f'(x) =
(ln((1/x^2)/(1/4x^2)))/(x*(ln (x^2))^2)`

`f'(x) = ln
4/(x*(ln (x^2))^2)`

Hence, the derivative of
the function is `f'(x) = ln 4/(x*(ln (x^2))^2).`