Q$1$: Solve the following problems $($each $10$ marks$)$

$1-$ Find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}$ of: $z=x^3-2xy{y}^2$

$2-$ Find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{y}}$ of: $z=cos3xsin4y$

$3-$ Find $\frac{de{l}^{2}z}{del{x}^2},\frac{de{l}^{2}z}{del{y}^2},\frac{de{l}^{2}z}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}}$ and $\frac{de{l}^{2}z}{\text{d}\text{e}\text{l}\text{y}\text{d}\text{e}\text{l}\text{x}}$ of: $z=2lnxy$

$4-$ The power $P$ consumed in a resistor is given by $P=\frac{V^2}{R}$ watts. Determine the approximate change in power when $V$ increases by $5\%$ and $R$ decreases by $0\mathrm{.}5\%$ if the original values of $V$ and $R$ are $50$ volts and $12\mathrm{.}5$ ohms respectively.

$5-$ Find the maxima, minima and saddle points for the following function:

$f(x,y)=x^2{y}^2-2x4y8$