use implicit differentiation to find dy/dx . Then evaluate the derivative function at the designated point.
x2y2 − 1= 0;(0.5, 2)
To find the derivative of y with respect to x (dy/dx) using implicit differentiation, we follow these steps:
1. Differentiate both sides of the equation with respect to x.
2. Apply the chain rule when necessary.
3. Solve the resulting equation for dy/dx.
4. Substitute the given point into the derivative function to evaluate it.
Let's follow these steps for the given equation, x^2y^2 - 1 = 0.
1. Differentiate both sides of the equation with respect to x:
d/dx(x^2y^2 - 1) = d/dx(0)
2xy^2 + 2x^2yy' - 0 = 0
2. Apply the chain rule to the terms involving y. The derivative of y^2 with respect to x is 2yy', where y' represents dy/dx:
2xy^2 + 2x^2(2yy') = 0
3. Solve the equation for dy/dx. Rearrange the terms to isolate dy/dx:
2xy^2 + 4x^2yy' = 0
4x^2yy' = -2xy^2
dy/dx = (-2xy^2) / (4x^2y)
= -y / (2x)
Now we have the derivative function, dy/dx = -y / (2x).
4. Evaluate the derivative function at the point (0.5, 2):
dy/dx = -2 / (2 * 0.5)
= -2 / 1
= -2
Therefore, the value of dy/dx at the point (0.5, 2) is -2.