Question

How do you implicitly differentiate `11=(x)/(1-ye^x)`?

Answer 1

Consider `f(x,y(x))=0` taking the derivative gives
`(df)/dx = (f_x) +(f_y) (dy)/dx = 0` so `(dy)/dx=-f_x/f_y`.
Now `f(x,y) = x/(1-ye^x)-11`
`f_x = d/dx(x/(1-ye^x)-11) = (1-e^x(x-1)y)/(e^x y-1)^2`
`f_y = d/dy(x/(1-ye^x)-11) = (y(x-1)-e^-x)/(e^x y-1)^2`
finally `(dy)/dx =(y(1-x)-e^-x)/x`