Final answer:
To find the implicit derivative of the given equation x² x²y 4y²=6, we differentiate both sides of the equation with respect to x. Using the product rule and chain rule, we obtain the implicit derivative dy/dx = (-2x - (2x)(y)) / (x² + 8y).
Step-by-step explanation:
The given equation is x² x²y 4y²=6. We need to find the implicit derivative for this equation. To do this, we differentiate both sides of the equation with respect to x. For the left-hand side, we use the product rule and chain rule. The derivative of x² with respect to x is 2x, and for x²y, the derivative is (2x)(y) + x²(dy/dx). Similarly, the derivative of 4y² is 8y(dy/dx).
Combining the derivatives of the left-hand side, we get 2x + (2x)(y) + x²(dy/dx) + 8y(dy/dx) = 0. To find the implicit derivative dy/dx, we isolate it on one side of the equation. Rearranging terms, we have x²(dy/dx) + 8y(dy/dx) = -2x - (2x)(y). Factoring out dy/dx, we get (x² + 8y)(dy/dx) = -2x - (2x)(y). Finally, we divide both sides by (x² + 8y) to get the implicit derivative: dy/dx = (-2x - (2x)(y)) / (x² + 8y).