Find $f_{x}, f_{y}, f_{x}(-4,1)$, and $f_{y}(-1,-7)$ for the following equation.
\[
f(x, y)=\sqrt{x^{2}+y^{2}}
\]

Step 1 ：Given the function \(f(x, y)=\sqrt{x^{2}+y^{2}}\), we are asked to find the partial derivatives of the function with respect to \(x\) and \(y\), and then to evaluate these at specific points.

Step 2 ：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 3 ：To find \(f_{x}\), we will differentiate \(f(x, y)\) with respect to \(x\), treating \(y\) as a constant. Similarly, to find \(f_{y}\), we will differentiate \(f(x, y)\) with respect to \(y\), treating \(x\) as a constant.

Step 4 ：The partial derivative of \(f(x, y)=\sqrt{x^{2}+y^{2}}\) with respect to \(x\) is \(f_{x} = \frac{x}{\sqrt{x^{2}+y^{2}}}\), and with respect to \(y\) is \(f_{y} = \frac{y}{\sqrt{x^{2}+y^{2}}}\).

Step 5 ：Evaluating these at the given points, we find \(f_{x}(-4,1) = \frac{-4}{\sqrt{17}}\) and \(f_{y}(-1,-7) = \frac{-7}{\sqrt{50}}\).

Step 6 ：So, the final answers are \(\boxed{f_{x} = \frac{x}{\sqrt{x^{2}+y^{2}}}\), \(\boxed{f_{y} = \frac{y}{\sqrt{x^{2}+y^{2}}}\), \(\boxed{f_{x}(-4,1) = \frac{-4}{\sqrt{17}}}\), and \(\boxed{f_{y}(-1,-7) = \frac{-7}{\sqrt{50}}}\).