We first find the first derivative, paying careful attention to #y(x)# where we need to apply the chain rule:
#d/(dx) ( x^3 + y^3 = 18xy)#
#rarr 3x^2 + 3y^2(dy)/(dx) = 18y + 18x(dy)/(dx)#
Rearrange to obtain expression for #(dy)/(dx)#:
#(dy)/(dx) = (3x^2 - 18y)/(18x - 3y^2)" " color(red)("Equation 1")#
Simplify:
#(dy)/(dx) = (x^2-6y)/(6x-y^2)#
Differentiate again using quotient rule:
#(d^2y)/(dx^2) = ((2x-6(dy)/(dx))(6x-y^2) - (x^2-6y)(6 - 2y(dy)/(dx)))/(6x-y^2)^2#
#(d^2y)/(dx^2) = (color(blue)(12x^2) - 2xy^2 -36x(dy)/(dx) + color(orange)(6y^2(dy)/(dx))- color(blue)( 6x^2) + 2x^2y(dy)/(dx) + 36y -color(orange)( 12y^2(dy)/(dx)))/(6x-y^2)^2#
Some stuff cancels which is nice.
#(d^2y)/(dx^2) = (6x^2 - 2xy^2 - 36x(dy)/(dx) - 6y^2(dy)/(dx) + 2x^2y(dy)/(dx) + 36y)/(6x-y^2)^2#
Now we want to sub in equation 1:
#= (6x^2 - 2xy^2 - 36x((x^2-6y)/(6x-y^2)) - 6y^2((x^2-6y)/(6x-y^2)) + 2x^2y((x^2-6y)/(6x-y^2)) + 36y)/(6x-y^2)^2#
Make every term in the numerator over a common denominator:
#= (6x^2(6x-y^2)/(6x-y^2) - 2xy^2(6x-y^2)/(6x-y^2) - 36x((x^2-6y)/(6x-y^2)) - 6y^2((x^2-6y)/(6x-y^2)) + 2x^2y((x^2-6y)/(6x-y^2)) + 36y(6x-y^2)/(6x-y^2))/(6x-y^2)^2#
Can move this down to the full denominator:
#=(6x^2(6x-y^2) - 2xy^2(6x-y^2) - 36x(x^2-6y) - 6y^2(x^2-6y) + 2x^2y(x^2-6y) + 36y(6x-y^2))/(6x-y^2)^3#
Expanding:
#=(color(blue)(36 x^3)-6 x^2 y^2-12 x^2 y^2+2 x y^4-color(blue)(36 x^3)+216 x y-6 x^2 y^2+color(orange)(36 y^3)+2 x^4 y-12 x^2 y^2+216 x y-color(orange)(36 y^3))/(6x-y^2)^3#
Again, we can cancel a few things, marked in colour.
We now collect like terms to obtain:
#=(2xy^4 + 432xy - 36x^2y^2 + 2x^4y)/(6x-y^2)^3#
Factor to finally obtain that:
#(d^2y)/(dx^2)=(2xy(y^3 + 216 - 18xy + x^3))/(6x-y^2)^3#