Final answer:
To solve the given equation using implicit differentiation, we apply the product rule to the term xy^2 and differentiate -2y + 8x with respect to x, then solve for dy/dx.
Step-by-step explanation:
To solve the equation xy^2 = -2y + 8x using implicit differentiation, we differentiate both sides of the equation with respect to x. This involves using the product rule as well as the chain rule for differentiation.
The product rule is given by d(uv)/dx = u(dv/dx) + v(du/dx), and the chain rule is d(f(g(x)))/dx = f'(g(x))g'(x).
Applying the product rule to the left-hand side with u = x and v = y^2, we get:
du/dx = 1 (since x is just x to the power of 1)
dv/dx = 2y(dy/dx) (since we are taking the derivative of y^2 with respect to x, hence the appearance of dy/dx by the chain rule)
So, the differentiation of the left side is x*(2y(dy/dx)) + y^2*(1).
For the right-hand side, we differentiate -2y + 8x with respect to x, which gives us:
d(-2y)/dx = -2(dy/dx) (since y is a function of x)
d(8x)/dx = 8 (since the derivative of x with respect to x is 1)
Setting both derivatives equal gives us:
x*2y(dy/dx) + y^2 = -2(dy/dx) + 8
To find dy/dx, we collect terms containing dy/dx on one side and factor out dy/dx:
(x*2y + 2)dy/dx = -y^2 + 8
Then, divide by (2xy + 2) to isolate dy/dx:
dy/dx = (-y^2 + 8)/(2xy + 2)
That's your implicit differentiation solution.