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What is the value of dx/dy at the point (1, 1) for the equation x^2y - y^2 =2?

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Final answer:

To find the value of dx/dy at the point (1, 1) for the equation x^2y - y^2 =2, we differentiate the equation implicitly with respect to x and y, substitute the given point back into the equation, and solve for dy/dx. The value of dx/dy at the point (1, 1) is -1.

Step-by-step explanation:

To find the value of dx/dy at the point (1, 1) for the equation x^2y - y^2 =2, we can differentiate the equation implicitly with respect to x and y. Differentiating implicitly, we get:

2xydx + x^2dy - 2yy'dx = 0

Now, we can substitute the point (1, 1) back into the equation and solve for dy/dx:

2(1)(1)dx + (1)^2 dy - 2(1)(dy/dx)dx = 0

2dx + dy - 2dy/dx dx = 0

2dx - 2dy/dx dx = -dy

(2 - 2dy/dx) dx = -dy

Now, we can rearrange and solve for dy/dx:

dy/dx = -dy / (2dx - dx)

dy/dx = -dy / dx(2 - 1)

dy/dx = -dy / dx

At the point (1, 1), dx and dy are both equal to 1. Therefore, dy/dx = -1 / 1 = -1.

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