Question

Find y'' by implicit differentiation. 4x3 + 5y3 = 2

Answer 1

Differentiate implicitly using the power rule: 

12x^2 + 15y^2y' = 0 

Make y' the subject: 

15y^2y' = -12x^2 
y' = -(4x^2) / 3y^2 

Differentiate again using the quotient rule: 

y'' = [-8x(3y^2) - 6yy'(-4x^2)] / 9y^4 
y'' = (-24xy^2 + 24x^2yy') / 9y^4 

Simplify: 

y'' = (-8xy + 8x^2y') / 3y^3 

Substitute (4x^2) / 3y^2 back for y': 

y'' = (-8xy + 8x^2 * 4x^2 / 3y^2) / 3y^3 
y'' = (-24xy^3 + 32x^4) / 9y^5