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 $\frac{\partial z}{\partial y} = \boxed{10}$.