use implicit differentiation to find dy/dx . Then evaluate the derivative function at the designated point
2xy2 – y = 10; (1, −2)
To find the derivative of a function using implicit differentiation, follow these steps:
Step 1: Write down the given equation.
2xy^2 – y = 10
Step 2: Differentiate both sides of the equation with respect to x. Treat y as a function of x and use the chain rule when necessary.
d/dx (2xy^2 – y) = d/dx (10)
Step 3: Apply the power rule and the chain rule to differentiate the terms on the left side of the equation.
2y^2 * d/dx (x) + 2x * d/dx (y^2) - d/dx (y) = 0
Step 4: Simplify the derivatives on the left side using the power rule.
2y^2 + 2x * (d/dx (y^2)) - d/dx (y) = 0
Step 5: Apply the chain rule to evaluate the derivative of y^2.
2y^2 + 2x * (2y * d/dx (y)) - d/dx (y) = 0
Step 6: Simplify the expression.
2y^2 + 4xy * (dy/dx) - dy/dx = 0
Step 7: Collect the terms involving dy/dx on one side of the equation.
4xy * (dy/dx) - dy/dx = -2y^2
Step 8: Factor out dy/dx.
dy/dx * (4xy - 1) = -2y^2
Step 9: Solve for dy/dx by dividing both sides by (4xy - 1).
dy/dx = -2y^2 / (4xy - 1)
Now, to evaluate dy/dx at the point (1, -2), substitute x = 1 and y = -2 into the derivative function obtained in Step 9.
dy/dx = -2(-2)^2 / (4(1)(-2) - 1)
dy/dx = -8 / (-8 - 1)
dy/dx = -8 / -9
dy/dx = 8/9
Therefore, the value of dy/dx at the point (1, -2) is 8/9.