Question

Question #36347

Answer 1

#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))))#

Explanation:

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))))#