Determine the derivative of the function f(x): $f(x)=2 x^{3} \ln (x)$

Step 1 ：Define the function \(f(x)=2 x^{3} \ln (x)\)

Step 2 ：Apply the product rule for differentiation, which states that the derivative of the product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function

Step 3 ：Calculate the derivative of \(x^{3}\) to get \(3x^{2}\)

Step 4 ：Calculate the derivative of \(\ln (x)\) to get \(1/x\)

Step 5 ：Substitute these derivatives back into the product rule to get \(f'(x)=2x^{3}*(1/x) + 2*3x^{2}*\ln(x)\)

Step 6 ：Simplify this to get \(f'(x)=2x^{2} + 6x^{2}\ln(x)\)

Step 7 ：Final Answer: The derivative of the function \(f(x)=2 x^{3} \ln (x)\) is \(f'(x)=\boxed{6x^2\ln(x) + 2x^2}\)