Consider the functions $z=9e^{x}lny,x=ln(ucosv),$ and $y=$usinv.

$($a$)$ Express $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{u}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{v}}$ as functions of $u$ and $v$ both by using the Chain Rule and by expressing $z$ directly in terms of $u$ and $v$ before differentiating.

$($b$)$ Evaluate $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{u}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{v}}$ at $(u,v)=(9,\frac{}{6})$

$($a$)$ Find each partial derivative needed to use the Chain Rule to find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{u}}.$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}=,\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{u}}=$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}=,\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{u}}=$

Express $z$ directly in terms of $u$ and $v.$

$z=$

$$

$($Use parentheses to clearly denote the argument of each function.$)$

Using either method, $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{u}}=$

$($Type an expression using $u$ and $v$ as the variables. Use parentheses to clearly denote the argument of each function.$)$

Find each partial derivative needed to use the Chain Rule to find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{v}}.$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}=$

$\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}=$

$$

$$

$\frac{\text{d}\text{e}\text{l}\text{x}}{\text{d}\text{e}\text{l}\text{v}}=$

$\frac{\text{d}\text{e}\text{l}\text{y}}{\text{d}\text{e}\text{l}\text{v}}=$

$$

$$