Problem #$3$: PARTIAL DERIVATIVES, LIMITS, CHAIN RULE...

A$)$ Evaluate the following limit or determine that it does not exist:

$\underset{(x,y)(0,0)}{lim}\frac{sin(xy)}{x^2+y^2}$

B$)$ Find the first partial derivatives of the following function:

$g(x,y,z)=\frac{xyz}{x+y}$

C$)$ Chain Rule: Use the chain rule to evaluate the following derivatives:

$w^{\text{'}}(t),$ where $w=$xysin$(z),x=t^2,y=4t^3$&$z=t+1.$

$w_s^{\text{'}}$ and $w_t^{\text{'}},$ where $w=xyz,x=2st,y=s{t}^2$&$z=s^{2}t.$

D$)$ Gradient and Directional Derivatives: Compute the gradient of the following function:

$f(x,y,z)=xy+yz+zx+4,$ evaluate it at the given point $P(2,-2,1)$ and evaluate the directional derivative at that point in the given direction:

vec$(u)=($:$0,-\frac{1}{\sqrt[\phantom{2}]{2}},-\frac{1}{\sqrt[\phantom{2}]{2}}$:$).$

$2$