Implicit Differentiation

Problem : 20) Find the second derivative of (d²y/dx²) ,
if (x³ - 3xy + y³ ) = 1


Solution : 


Differentiating, we get  3x² - 3x * dy/dx + 3y² * dy/dx ) = 0
=> dy/dx = ( y - x²) / (y² - x) - - - - - (1)

Now, note that:
(i) (y² - x) * (dy/dx) = ( y - x²) ;
(ii) ( y - x²) * (dy/dx) = (y - x²)² / (y² - x)
(iii) 3x²y - x⁴ - xy³ = -x (x³ - 3xy + y³ ) ==> -x

Differentiating (1) , we get,

[(y² - x) * (dy/dx - 2x) - (y - x²) * (2y * (dy/dx) - 1) ] / (y² - x)²
=> [(y² - x) * (dy/dx) - 2x * (y² - x) - 2y * (y - x²) (dy/dx) + (y - x²)] / (y² - x)²

Using (i) & (ii) ,
. . [(y - x²) - 2x *(y² - x) - 2y * {(y - x²)² / (y² - x)} + y - x² ] / (y² - x)²
. . [ 2y - 2x² - 2xy² + 2x² - 2y * {(y - x²)² / (y² - x)} ] / (y² - x)²

. . 2y [ (1 - xy) * (y² - x) - {y² + x⁴ - 2x²y} ] /(y² - x)³
. . 2y [y² - x - xy³ + x²y - y² - x⁴ + 2x²y ] /(y² - x)³
. .2y [ - x - xy³ + x²y - x⁴ + 2x²y ] /(y² - x)³
. . 2y [- x + (-x) * (x³ - 3xy + y³ ) ] /(y² - x)³

Using (iii) we have,
==> 2y * (-2x) /(y² - x)³
==> -4xy / (y² - x)³ ==>(Answer)