Calculus question implicit derivatives???

Question

Determine whether the given relation is an implicit solution to the given differential equation using implicit differentiation.

e^xy + y = x - 1, dy/dx = (e^-xy) -(y) / (e^-xy) + (x) ....... I know its yes but I cant verify

Answer 1

Well, if you calculate the derivative of both sides of your diff. eq. you'll get:

d(e^xy)/dx+dy/dx=dx/dx

(d(xy)/dx)e^xy+dy/dx=1

(xdy/dx+y)e^xy+dy/dx=1

(xe^xy+1)dy/dx=1-ye^xy

multiply both sides of the equation by e^-xy :

(x+e^-xy)dy/dx=e^-xy-y

Thus:

dy/dx=(e^-xy-y)/(e^-xy+x)

Voilà!