# Question 36347

Aug 19, 2016

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 $f \left(x , y\right) = 0$ then

$\mathrm{df} = {f}_{x} \mathrm{dx} + {f}_{y} \mathrm{dy} = 0$

so

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {f}_{x} / {f}_{y}$

If $f \left(x , y\right) = 3 y - \cot \left(y {e}^{\tan x} / {\log}_{e} x\right)$ 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) )

${f}_{y} = 3 + \frac{{e}^{T} a n \left(x\right) C s c {\left(\frac{{e}^{T} a n \left(x\right) y}{L} o {g}_{e} \left(x\right)\right)}^{2}}{L} o {g}_{e} \left(x\right)$

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))))#