You just need to make use of the quotient rule and the chain rule.
Quotient Rule:
d/dx[f(x)/g(x)]=(g(x)f'(x)-f(x)g'(x))/(g(x))^2
Chain Rule:
d/dx[f(g(x))]=f'(g(x))*g'(x)
(basically, you get the derivative of f'(x) then multiply by the derivative of g(x))
Solution:
d/dx[ln(2x)/tanx]
By quotient rule:
[1]" "=(tanx*D_x[ln(2x)]-ln(2x)*D_x[tanx])/tan^2x
The derivative of tanx is sec^2x:
[2]" "=(tanx*D_x[ln(2x)]-ln(2x)*sec^2x)/tan^2x
To get the derivative of ln(2x), you use chain rule. The derivative of ln(x) is 1/x
[3]" "=(tanx*1/(2x)*D_x[2x]-ln(2x)*sec^2x)/tan^2x
[4]" "=(tanx*1/(2x)*2-ln(2x)*sec^2x)/tan^2x
[5]" "=(tanx*1/x-ln(2x)*sec^2x)/tan^2x*x/x
[6]" "=color(red)((tanx-xln(2x)sec^2x)/(xtan^2x))