Final answer:
To find dy/dx for xey = x - y, implicit differentiation is applied, and the product and chain rules are used to derive dy/dx = (1 - ey)/(xey + 1).
Step-by-step explanation:
To find dy/dx by implicit differentiation for the given equation xey = x - y, we differentially both sides of the equation with respect to x.
First, we apply the product rule to the left side, which states that d(uv)/dx = u(dv/dx) + v(du/dx). Our u is x and our v is ey. Since ey is a function of y, and y is a function of x, we will also need to apply the chain rule. This gives us:
- x(d/dx)(ey) + ey(d/dx)(x) = x(ey)(dy/dx) + ey
On the right side, we simply differentiate each term normally with respect to x, remembering to apply the chain rule to -y to get -(dy/dx). This gives:
- d/dx(x - y) = 1 - (dy/dx)
Equating both sides, we get:
x(ey)(dy/dx) + ey = 1 - (dy/dx)
To solve for dy/dx, we gather all dy/dx terms on one side and the other terms on the opposite side:
x(ey)(dy/dx) + (dy/dx) = 1 - ey
Factoring out dy/dx, we have:
(dy/dx)(x(ey) + 1) = 1 - ey
Finally, we divide both sides by (x(ey) + 1):
dy/dx = (1 - ey)/(x(ey) + 1)
This is the derivative of y with respect to x for the given implicit function.