Final answer:
To find dy/dx by implicit differentiation, differentiate both sides of the equation x² + xy = y²/2 with respect to x. Use the product rule and the chain rule to isolate dy/dx and solve for it.
Step-by-step explanation:
To find dy/dx by implicit differentiation for the equation x² + xy = y²/2, we can differentiate both sides of the equation with respect to x.
Using the product rule for differentiation on the left-hand side (LHS) and the chain rule for differentiation on the right-hand side (RHS), we get:
2x + x(dy/dx) = (2y/2)(dy/dx)
Now, we can isolate dy/dx by moving all the terms involving it to one side:
x(dy/dx) - (y/2)(dy/dx) = -2x
Factoring out dy/dx, we have:
(x - y/2)(dy/dx) = -2x
Finally, we can solve for dy/dx by dividing both sides by (x - y/2):
dy/dx = (-2x) / (x - y/2)