Question

Given y^4 -x^4=16 find and simplify d2y/dx2 using implicit differentiation.
I got the first derivative which was x^3/y^3 and no im am stuck on using implicit differentiation for the second derivative part

Answer 1

y^4 - x^4 = 16
4y^3 y' - 4x^3 = 0
y' = x^3/y^3

y'' = (3x^2*y^3 - x^3*3y^2*y')/y^6
= 3x^2(y-xy')/y^4
= 3x^2 (y-x*x^3/y^3)/y^4
= 3x^2 (y^4 - x^4)/y^7
= 48x^2/y^7

or,

y^3 y' = x^3
3y^2 y'^2 + y^3 y'' = 3x^2
3y^2 (x^6/y^6) + y^3 y'' = 3x^2

y^3 y'' = 3x^2 - 3x^6/y^4
y'' = 3x^2 (y^4 - x^4)/y^7