Step 1 :The question is asking for the partial derivatives of the function \(f(x, y, z)=4 x y+7 x z+y z\) with respect to \(x\), \(y\), and \(z\).
Step 2 :Partial derivatives are a type of derivative where we differentiate with respect to one variable while keeping the other variables constant.
Step 3 :To find \(f_{x}\), we differentiate \(f\) with respect to \(x\) while treating \(y\) and \(z\) as constants.
Step 4 :To find \(f_{y}\), we differentiate \(f\) with respect to \(y\) while treating \(x\) and \(z\) as constants.
Step 5 :To find \(f_{z}\), we differentiate \(f\) with respect to \(z\) while treating \(x\) and \(y\) as constants.
Step 6 :\(f = 4*x*y + 7*x*z + y*z\)
Step 7 :\(f_x = 4*y + 7*z\)
Step 8 :\(f_y = 4*x + z\)
Step 9 :\(f_z = 7*x + y\)
Step 10 :Final Answer: \(f_{x}=\boxed{4y+7z}\), \(f_{y}=\boxed{4x+z}\), \(f_{z}=\boxed{7x+y}\)