Question

Question #47835

Answer 1

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)))`