Problem $5$: $(6$ pts$)$: Let's consider a function $f(x,y)$ of two variables defined by:

$f(x,y)=x{y}^2+4xy+y^5$

We are asked to find the four second$-$order partial derivatives which are: $\frac{de{l}^{2}f}{del{x}^2},\frac{de{l}^{2}f}{del{y}^2},\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{x}\text{d}\text{e}\text{l}\text{y}},$ and $\frac{de{l}^{2}f}{\text{d}\text{e}\text{l}\text{y}\text{d}\text{e}\text{l}\text{x}}.$