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))#
