How do I find the value of d^2y/dx^2 for the function defined implicitly by xy^2 + y = 2 at the point (1,-2)?

xy^2+y=2

y^2 dx + 2xy dy+dy=0
dy/dx ( 2xy+1)=y^2
dy/dx= y^2(1/(2xy+1)

dy"/dx"= 2y/(2xy+1) -y^2/(2xy+1)^2 * (2y*xdy/dx+2y)

Put for dy/dx y^2/(2xy+1)

then put number (1,-2)

CHECK MY WORK, I have a headache (Elm pollen).

To find the value of d^2y/dx^2 for the given implicitly defined function at a specific point, you can follow these steps:

Step 1: Differentiate the given implicit function implicitly with respect to x. This will involve using both the product rule and the chain rule.

Differentiating xy^2 + y = 2 implicitly with respect to x:
(x*d(y^2)/dx) + (y*2y*dx/dx) + dy/dx = 0

Simplifying the above expression:
x*(2y*(dy/dx)) + 2y + dy/dx = 0

Step 2: Now, we have obtained an equation involving dy/dx. We can solve this equation for dy/dx using algebraic manipulation.

Rearranging the equation and factoring out dy/dx:
2xy*(dy/dx) + dy/dx = -2y

Combining like terms:
(2xy + 1) * (dy/dx) = -2y

Dividing both sides by (2xy + 1):
dy/dx = -2y / (2xy + 1)

Step 3: Evaluate dy/dx at the given point (1, -2) to find the slope of the tangent line at that point.

Substituting x = 1 and y = -2 into the expression for dy/dx:
dy/dx = -2(-2) / (2 * 1 * (-2) + 1)
= 4 / (-4 + 1)
= 4 / (-3)
= -4/3

Therefore, the slope of the tangent line at the point (1, -2) is -4/3.

Step 4: Differentiate the expression for dy/dx (which you obtained earlier) with respect to x using the quotient rule to find d^2y/dx^2.

Using the quotient rule to differentiate dy/dx = -2y / (2xy + 1) with respect to x:
d(dy/dx) / dx = [(0)(2xy + 1) - (-2y)(2x)] / (2xy + 1)^2

Simplifying the above expression:
d(dy/dx) / dx = 4yx / (2xy + 1)^2

Step 5: Finally, substitute x = 1 and y = -2 into the expression for d(dy/dx) / dx to find the value of d^2y/dx^2 at the given point (1, -2).

Substituting x = 1 and y = -2 into the expression:
d(dy/dx) / dx = 4(1)(-2) / (2(1)(-2) + 1)^2
= -8 / (-4 + 1)^2
= -8 / (-3)^2
= -8 / 9

Therefore, the value of d^2y/dx^2 for the given function at the point (1, -2) is -8/9.