Final answer:
To find dy/dx by implicit differentiation, we differentiate both sides of the equation with respect to x, treating y as an implied function of x. Applying the product rule and simplifying the equation, we find that dy/dx = (1 - (e^y))/(1 + x(e^y)).
Step-by-step explanation:
To find dy/dx by implicit differentiation, we differentiate both sides of the equation with respect to x, treating y as an implied function of x.
Using the product rule, we have:
(1) d(xe^y)/dx = d(x-y)/dx
Applying the product rule, we get:
x(d(e^y))/dx + (e^y)(dx/dx) = (1)(dx/dx) + (-1)(dy/dx)
Next, we simplify the equation and solve for dy/dx:
x(e^y)(dy/dx) + (e^y) = 1 - dy/dx
dy/dx + x(e^y)(dy/dx) = 1 - (e^y)
dy/dx(1 + x(e^y)) = 1 - (e^y)
dy/dx = (1 - (e^y))/(1 + x(e^y))