#4x^2 + 3y^2 = 6#
#d/(dx)(4x^2 + 3y^2) = d/(dx)(6)#
#8x + 6y(dy)/(dx) = 0#
So #(dy)/(dx) = (-4x)/(3y)#
On to the second derivative:
#d/(dx)((dy)/(dx)) =d/(dx) ( (-4x)/(3y))#
#(d^2y)/(dx^2) = ((-4)(3y)- (-4x)(3(dy)/dx))/((3y)^2)#
#(d^2y)/(dx^2) = (-12y +12 x(dy)/dx)/(9y^2)#
Replacing #dy/dx# by the expression above gives us:
#(d^2y)/(dx^2) = (-12y +12 x((-4x)/(3y)))/(9y^2) = (-12y + (-16x^2)/y)/(9y^2)#
No multiply by #y/y# to get
#(d^2y)/(dx^2) = (-12y^2 -16x^2)/(9y^3) = (-4(4x^2+3y^2))/(9y^3)#
Now, use the initial equation: #4x^2 + 3y^2 = 6# to write the answer as:
#(d^2y)/(dx^2) = = (-24)/(9y^3) = (-8)/(3y^3)#
