#y^2 = P#
What follows is mostly chain and product rules with implicit differentiation.
First derivative wrt #x#:
#2 y y_x = P_x#
Second derivative wrt #x#:
#2 (y_x)^2 + 2 y y\_( x x ) = P\_(x x)#
Third derivative wrt #x#:
#4 y_x y _(x x) + 2 y_x y\_( x x ) + 2 y y\_( x x x )= P\_(x x x)#
#implies 6 y_x y _(x x) + 2 y y\_( x x x )= P\_(x x x)#
#implies P\_(x x x) = color(red)(2 ( 3 y_x y _(x x) + y y\_( x x x )) )#
Now we have:
#2d/dx (y^3 y_(x x) ) = 3 y^2 y_x y _(x x) + y^3 y\_(x x x)#
#=y^2 * color(red)(2 (3 y_x y _(x x) + y y\_(x x x) ))#
#= P * P_(x x x)#
