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.