Find d2y/dx2 by implicit differentiation.

Question

Find d2y/dx2 by implicit differentiation.

Given:

x³ + y³ = 4

Answer 1

3x^2 + 3y^2 y' = 0

x^2 + y^2 y' = 0

y' = -x^2/y^2

y'' = [(y^2)(-2x) - (-x^2)(2yy')]/y^4 <--- quotient rule

y'' = [ (x^2)(2yy')-(y^2)(2x)]/y^4 <--- resolves the negatives

= 2xy [(xy') - y] / y^4

then you substitute y' = -x^2/y^2 into that

y'' = (2xy) [ (-x^3)/y^2 - y]/y^4

= (-2xy) [ y^3 + x^3] / y^6 <--- common denominator is y^2; factors out the negative