Given #f(x,y)=0# then
#df = f_x dx + f_y dy = 0#
so
#dy/dx = -f_x/f_y#
If #f(x,y) =3y- cot(y e^{tanx}/log_e x)# then
#f_x = Csc((e^Tan(x) y)/
Log_e(x))^2 ( (e^Tan(x) y Sec(x)^2)/
Log_e(x)-(e^Tan(x) y)/(x Log_e(x)^2) )#
#f_y = 3 + (e^Tan(x) Csc((e^Tan(x) y)/Log_e(x))^2)/Log_e(x)#
then
#dy/dx =- ( Csc((e^Tan(x) y)/
Log_e(x))^2 ( (e^Tan(x) y Sec(x)^2)/
Log_e(x)-(e^Tan(x) y)/(x Log_e(x)^2) ))/(3 + (e^Tan(x) Csc((e^Tan(x) y)/Log_e(x))^2)/Log_e(x))#
or simplifying
#dy/dx = -((e^Tan(x) y Csc((e^Tan(x) y)/Log_e(x))^2 (x Log_e(x) Sec(x)^2)-1)/(
x Log_e(x) (e^Tan(x)Csc((e^Tan(x) y)/Log_e(x))^2 + 3 Log_e(x))))#