Let u=x2-3x, then ∂u/∂x=2x-3, ∂u/∂y=∂2u/∂y2=0;
Let v=(1+x2y2)-n, then ∂v/∂x=-n(2xy2)(1+x2y2)-n-1, ∂v/∂y=-n(2x2y)(1+x2y2)-n-1.
These derivatives may be useful as we progress through the 1st and 2nd derivatives:
∂f/∂x, ∂f/∂y, (∂/∂x)(∂f/∂y), (∂/∂y)(∂f/∂x), ∂2f/∂x2, ∂2f/∂y2.
We would expect the 3rd and 4th of these listed derivatives to be equal.
∂f/∂x=∂(uv)/∂x=v∂u/∂x+u∂v/∂x=(1+x2y2)-1(2x-3)-(x2-3x)(2xy2)(1+x2y2)-2,
∂f/∂x=(1+x2y2)-1(2x-3)-(2x3y2-6x2y2)(1+x2y2)-2,
∂f/∂x=[(2x-3)(1+x2y2)-(2x3y2-6x2y2)](1+x2y2)-2,
∂f/∂x=(2x+2x3y2-3-3x2y2-2x3y2+6x2y2)(1+x2y2)-2,
∂f/∂x=(3x2y2+2x-3)(1+x2y2)-2.
Let u=Let u=x2-3x, then ∂u/∂x=2x-3, ∂u/∂y=∂2u/∂y2=0; ∂2u/∂x2=2.
Let v=(1+x2y2)-n, then ∂v/∂x=-n(2xy2)(1+x2y2)-n-1, ∂v/∂y=-n(2x2y)(1+x2y2)-n-1.
These derivatives may be useful as we progress through the 1st and 2nd derivatives:
∂f/∂x, ∂f/∂y, (∂/∂x)(∂f/∂y), (∂/∂y)(∂f/∂x), ∂2f/∂x2, ∂2f/∂y2.
We would expect the 3rd and 4th of these listed derivatives to be equal.
∂f/∂x=∂(uv)/∂x=v∂u/∂x+u∂v/∂x=(1+x2y2)-1(2x-3)-(x2-3x)(2xy2)(1+x2y2)-2,
∂f/∂x=(1+x2y2)-1(2x-3)-(2x3y2-6x2y2)(1+x2y2)-2,
∂f/∂x=[(2x-3)(1+x2y2)-(2x3y2-6x2y2)](1+x2y2)-2,
∂f/∂x=(2x+2x3y2-3-3x2y2-2x3y2+6x2y2)(1+x2y2)-2,
∂f/∂x=(3x2y2+2x-3)(1+x2y2)-2.
∂f/∂y=∂(uv)/∂y=v∂u/∂y+u∂v/∂y=0-(2x2y)(x2-3x)(1+x2y2)-2=(-2x4y+6x3y)(1+x2y2)-2.
Let u=3x2y2+2x-3, then ∂u/∂y=6x2y; v=(1+x2y2)-2.
(∂/∂y)(∂f/∂x)=∂(uv)/∂y=v∂u/∂y+u∂v/∂y,
(∂/∂y)(∂f/∂x)=6x2y(1+x2y2)-2+(3x2y2+2x-3)(-2(2x2y)(1+x2y2)-3,
(∂/∂y)(∂f/∂x)=[(6x2y(1+x2y2)-4x2y(3x2y2+2x-3)](1+x2y2)-3,
(∂/∂y)(∂f/∂x)=(6x2y+6x4y3-12x4y3-8x3y+12x2y)(1+x2y2)-3,
(∂/∂y)(∂f/∂x)=(-6x4y3-8x3y+18x2y)(1+x2y2)-3.
∂f/∂y=(-2x4y+6x3y)(1+x2y2)-2; let u=-2x4y+6x3y, then ∂u/∂x=-8x3y+18x2y; v=(1+x2y2)-2;
(∂/∂x)(∂f/∂y)=∂(uv)/∂x=v∂u/∂x+u∂v/∂x,
(∂/∂x)(∂f/∂y)=(1+x2y2)-2(-8x3y+18x2y)+(-2x4y+6x3y)(-2(2xy2)(1+x2y2)-3,
(∂/∂x)(∂f/∂y)=[(-8x3y+18x2y)(1+x2y2)-4xy2(-2x4y+6x3y)](1+x2y2)-3,
(∂/∂x)(∂f/∂y)=(-8x3y+18x2y-8x5y3+18x4y3+8x5y3-24x4y3)(1+x2y2)-3,
(∂/∂x)(∂f/∂y)=(-6x4y3-8x3y+18x2y)(1+x2y2)-3=(∂/∂y)(∂f/∂x), confirming expectation.
Let u=3x2y2+2x-3, then ∂u/∂x=6xy2+2; v=(1+x2y2)-2.
∂2f/∂x2=v∂u/∂x+u∂v/∂x=(6xy2+2)(1+x2y2)-2-4xy2(3x2y2+2x-3)(1+x2y2)-3,
∂2f/∂x2=[(6xy2+2)(1+x2y2)-4xy2(3x2y2+2x-3)](1+x2y2)-3,
∂2f/∂x2=(6xy2+2+6x3y4+2x2y2-12x3y4-8x2y2+12xy2)(1+x2y2)-3,
∂2f/∂x2=(-6x3y4-6x2y2+18xy2+2)(1+x2y2)-3.
Let u=-2x4y+6x3y, then ∂u/∂y=-2x4+6x3; v=(1+x2y2)-2.
∂2f/∂y2=v∂u/∂y+u∂v/∂y=(1+x2y2)-2(-2x4+6x3)-4x2y(-2x4y+6x3y)(1+x2y2)-3,
∂2f/∂y2=[(1+x2y2)(-2x4+6x3)-4x2y(-2x4y+6x3y)](1+x2y2)-3,
∂2f/∂y2=(-2x4+6x3-2x6y2+6x5y2+8x6y2-24x5y2)(1+x2y2)-3,
∂2f/∂y2=(6x6y2-18x5y2-2x4+6x3)(1+x2y2)-3.