Let $fbe$ a function defined $as$ follows:

$f(x,y)=\{\begin{array}{ccc}\frac{9x^{3}y}{10x^2+11y^2},& \text{;}(x,y)(0,& 0)\\ 0,& \text{;}(x,y)=(0,& 0)\end{array}$

$i.$ Find $(\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}{)}_{(0,0)}.$

$ii.$ Find $(\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}})$ for $(x,y)(0,0).$

iii. Find $(\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{y}\text{d}\text{e}\text{l}\text{x}}{)}_{(0,0)}.$

$iv.$ Find $(\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}{)}_{(0,0)}.$

$v.$ Find $(\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}{)}_{(0,0)}$ for $(x,y)(0,0).$

$vi.$ Find $(\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}{)}_{(0,0)}.$

vii. State the relationship between $(\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}{)}_{(0,0)}$ and $(\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{y}\text{d}\text{e}\text{l}\text{x}}{)}_{(0,0)}.$