The quotient rule states that, given an expression #f(x) = g(x)/(h(x))#, the derivative #f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2#
In the case above, our functions are:
#g(x) = ln(9x)# and #h(x) = 1+x#.
Giving us derivatives of:
#g'(x) = (d/dx)ln(9x) = 9/(9x) = 1/x# and #h'(x) = (d/dx)(1+x) = 1#
Thus, by the Quotient Rule:
#f'(x)= ((1/x)(1+x) - (ln(9x))(1))/(1+x)^2 = (1/x + 1 - ln (9x))/(x^2+2x+1) #
Note that #g(x)# is undefined in the real number plane for any #x<=0#; therefore, the derivative will be undefined as well (recall that #ln(x)# does not have a real value for #x<=0#). Thus, since the only point at which our denominator is 0 would be #x=-1#, we need not worry about the denominator causing discontinuities within the domain of our function #f(x)#.
