Find y" by implicit differentiation.
x^3+y^3 = 1
(x^3)'+(y^3)' = (1)'
3x^2+3y^2(y') = 0
3y^2(y') = -3x^2
y' = -3x^2/3y^2
y' = -x^2/y^2
y" = [(y^2)(-x^2)'-(-x^2)(y^2)']/y^4
y" = [(y^2)(-2x)-(-x^2)(2y)]/y^4
y" = [(-2xy^2)-(-2yx^2)]/y^4
y" = -2xy(y-x)/y^4
y" = [-2x(y-x)]/y^3
My book shows the answer is y" = -2x/y^5.
Your second line should be (via the chain rule!)
y" = [(y^2)(-2x)-(-x^2)(2yy')]/y^4
Now plug in -x^2/y^2 for y' and you should get the right result.
x^3+y^3 = 1
(x^3)'+(y^3)' = (1)'
3x^2+3y^2(y') = 0
3y^2(y') = -3x^2
y' = -3x^2/3y^2
y' = -x^2/y^2
GOOD
y" = [(y^2)(-x^2)'-(-x^2)(y^2)']/y^4
then I disagree with next line
y" = [(y^2)(-2x)-(-x^2)(2y DY/DX)]/y^4
WHERE dy/dx = -x^2/y^2
So chain rule is necessary since x and y are different variables on one side, forming a composition?
the chain rule is always necessary. It just happen s that dx/dx = 1, so it doesn't enter into the complexity.
Suppose you had
x^3 + y^3 + u^2 = uv^2
Then you'd have
3x^2 + 3y^2 y' + 2u u' = u' v^2 = 2uv v'
Whenever you take a derivative, you have to factor in the chain rule.
It might be the case that x and y are both functions of t. Then you have
x^2 + y^2 = 10
2x dx/dt + 2y dy/dt = 0
To find y", the second derivative, by implicit differentiation, you first differentiate both sides of the equation with respect to x. Let's go through the steps again to identify where the discrepancy might be:
Given equation: x^3 + y^3 = 1
Differentiating both sides with respect to x:
(d/dx) [x^3 + y^3] = (d/dx) [1]
3x^2 + 3y^2 (dy/dx) = 0
Now, to find dy/dx, isolate it:
3y^2 (dy/dx) = -3x^2
dy/dx = -3x^2 / 3y^2
dy/dx = -x^2 / y^2
Now, to find y", the second derivative, we differentiate dy/dx with respect to x:
(d/dx) [dy/dx] = (d/dx) [-x^2 / y^2]
(d^2y/dx^2) = [(y^2) (-2x) - (-x^2) (2y)] / (y^2)^2
(d^2y/dx^2) = [(-2xy^2) + (2yx^2)] / y^4
(d^2y/dx^2) = (2yx^2 - 2xy^2) / y^4
(d^2y/dx^2) = -2x(y^2 - yx) / y^4
(d^2y/dx^2) = -2x(y - x) / y^3
From the above calculations, I believe the second derivative dy^2/dx^2 should be -2x(y - x) / y^3, not -2x/y^5 as mentioned in your book. Please double-check your book or consult with your instructor for clarification.