find dy/dx implicitly, given x^3+y^3=3xy
3 x^2 dx + 3 y^2 dy = 3 x dy + 3 y dx
(y^2 -x)dy = -(x^2-y)dx
dy/dx = - (x^2-y)/(y^2-x)
ty!
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To find the derivative, dy/dx, implicitly given the equation x^3 + y^3 = 3xy, we can use the implicit differentiation technique. This method allows us to differentiate both sides of the equation with respect to x while treating y as an implicit function of x.
Let's go step by step to find dy/dx:
1. Start with the given equation: x^3 + y^3 = 3xy.
2. Differentiate both sides of the equation with respect to x. For simplicity, let's consider each term on its own.
- For the first term, x^3, the derivative with respect to x is 3x^2 (using the power rule).
- For the second term, y^3, we need to use the chain rule since y is a function of x. The derivative is therefore 3y^2 * dy/dx.
- For the third term, 3xy, we use the product rule. The derivative is 3y + 3x * dy/dx.
3. Combine the terms: 3x^2 + 3y^2 * dy/dx = 3y + 3x * dy/dx.
4. Rearrange the equation to isolate dy/dx terms on one side: 3y^2 * dy/dx - 3x * dy/dx = 3y - 3x^2.
5. Factor out dy/dx: (3y^2 - 3x) * dy/dx = 3y - 3x^2.
6. Finally, divide both sides of the equation by (3y^2 - 3x) to solve for dy/dx:
dy/dx = (3y - 3x^2) / (3y^2 - 3x).
Therefore, the implicit derivative dy/dx of the equation x^3 + y^3 = 3xy is (3y - 3x^2) / (3y^2 - 3x).