Find the first partial derivatives of the following function. \[ \begin{array}{l} f(x, y)=x^{5}-3 x y-8 y^{2} \\ f_{x}(x, y)=\square \\ f_{y}(x, y)=\square \end{array} \]

Step 1 ：The question is asking for the first partial derivatives of the function with respect to x and y. The partial derivative of a function with respect to a variable is the derivative of the function with respect to that variable, treating all other variables as constants.

Step 2 ：For \(f_{x}(x, y)\), we will differentiate \(f(x, y)\) with respect to x, treating y as a constant.

Step 3 ：For \(f_{y}(x, y)\), we will differentiate \(f(x, y)\) with respect to y, treating x as a constant.

Step 4 ：The first partial derivative of the function with respect to x is \(f_{x}(x, y)=5x^{4}-3y\).

Step 5 ：The first partial derivative of the function with respect to y is \(f_{y}(x, y)=-3x-16y\).

Step 6 ：Final Answer: The first partial derivatives of the function are: \(\boxed{f_{x}(x, y)=5x^{4}-3y}\) and \(\boxed{f_{y}(x, y)=-3x-16y}\).