#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)#