Final answer:
To find dy/dx for the lemniscate, we use implicit differentiation and solve for dy/dx to get the result dy/dx = ⁄[2x - 4x(x² + y²)][4y(x² + y²) + 2y].
Step-by-step explanation:
To find dy/dx for the lemniscate given by the equation (x² + y²)² = x² - y², we apply implicit differentiation with respect to x. Differentiating each side of the equation, we get 2(x² + y²)(2x + 2ydy/dx) = 2x - 2ydy/dx. Now, solving for dy/dx, we combine like terms and isolate dy/dx.
Let's carry out the differentiation:
- 2(x² + y²)(2x + 2ydy/dx) = 2x - 2ydy/dx,
- 4x(x² + y²) + 4y(x² + y²)dy/dx = 2x - 2ydy/dx,
- 4y(x² + y²)dy/dx + 2ydy/dx = 2x - 4x(x² + y²),
- dy/dx(4y(x² + y²) + 2y) = 2x - 4x(x² + y²).
Finally, solving for dy/dx:
dy/dx = ⁄[2x - 4x(x² + y²)][4y(x² + y²) + 2y].