For the function $z=9 x^{3}+y^{2}+4 x y$, find $\frac{\partial z}{\partial x}, \frac{\partial z}{\partial y}, \frac{\partial}{\partial x} z(5,-5)$, and $\frac{\partial}{\partial y} z(5,-5)$

Step 1 ：We are given the function $z=9 x^{3}+y^{2}+4 x y$ and asked to find the partial derivatives with respect to $x$ and $y$, and then evaluate these at the point $(5,-5)$.

Step 2 ：To find the partial derivative of a function with respect to a variable, we differentiate the function with respect to that variable, treating all other variables as constants.

Step 3 ：For $\frac{\partial z}{\partial x}$, we differentiate $z$ with respect to $x$, treating $y$ as a constant. This gives us $\frac{\partial z}{\partial x} = 27x^{2} + 4y$.

Step 4 ：For $\frac{\partial z}{\partial y}$, we differentiate $z$ with respect to $y$, treating $x$ as a constant. This gives us $\frac{\partial z}{\partial y} = 4x + 2y$.

Step 5 ：To find the value of these partial derivatives at the point $(5,-5)$, we substitute $x=5$ and $y=-5$ into the partial derivatives.

Step 6 ：Substituting into $\frac{\partial z}{\partial x}$, we get $\frac{\partial z}{\partial x} = 655$ at the point $(5,-5)$.

Step 7 ：Substituting into $\frac{\partial z}{\partial y}$, we get $\frac{\partial z}{\partial y} = 10$ at the point $(5,-5)$.

Step 8 ：So, the partial derivative of $z$ with respect to $x$ is $\boxed{27x^{2} + 4y}$, the partial derivative of $z$ with respect to $y$ is $\boxed{4x + 2y}$. At the point $(5,-5)$, $\frac{\partial z}{\partial x} = \boxed{655}$ and $\frac{\partial z}{\partial y} = \boxed{10}$.