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.