Question #47835

1 Answer
Mar 15, 2017

See answer in explanation below

Explanation:

First we notice that this is a quotient so logically we must use the quotient rule to solve this. The quotient rule generically looks like

#f(x)/g(x) = ((g(x)f'(x) - f(x)g'(x))/(g(x)^2))#

And in this case we would use #ln(6x) = f(x) and ln(2x) = g(x)#

So to solve this equation we would want to find the derivative of both the top and the bottom

#f'(x) = 6/(6x) = 1/x and g'(x) = 2/(2x) = 1/x# and plug everything into the quotient rule would look like the following

#((ln(2x)(1/x)) - (ln(6x)(1/x)))/(ln(2x)^2)#

Which we can simplify to

#(((ln(2x)-ln(6x))/x)/(ln(2x)^2))#

Simplified even further looks like

#((ln(2x)-ln(6x))/(x(ln(2x)^2)))#