Find an explicit expression for y' for the curve de�fined by 3(x^2 + y^2)^2 = 100xy.

To find the explicit expression for y' (the derivative of y with respect to x) for the curve defined by the equation 3(x^2 + y^2)^2 = 100xy, we need to differentiate both sides of the equation with respect to x.

Differentiating the left side of the equation involves applying the chain rule. Let's break it down step by step:

1. Start with the equation: 3(x^2 + y^2)^2 = 100xy.
2. Differentiate both sides with respect to x:

d/dx [3(x^2 + y^2)^2] = d/dx [100xy]

3. Apply the chain rule to the left side of the equation. Let's call the expression (x^2 + y^2) as u:

d/du [u^2] * d/dx [x^2 + y^2] = 100y + 100x * dy/dx

Note: d/du [u^2] = 2u.

4. Differentiate x^2 + y^2 with respect to x. Since y is a function of x, we need to use the chain rule again:

d/dx [x^2] + d/dx [y^2] * dy/dx = 100y + 100x * dy/dx

Note: d/dx [x^2] = 2x.

5. Differentiate y^2 with respect to x using the chain rule:

2x + 2y * dy/dx = 100y + 100x * dy/dx

6. Rearrange the equation to solve for dy/dx:

2x - 100x * dy/dx = 100y - 2y * dy/dx

(2x - 100x) * dy/dx = 100y - 2y

(2 - 100) * x * dy/dx = 98y

-98x * dy/dx = 98y

dy/dx = -y/x

7. Therefore, the explicit expression for y' is dy/dx = -y/x.