Find the derivative of the function. \[ f(x)=\left(4 x^{2}-x\right) \ln (x) \] \[ f^{\prime}(x)= \]

Step 1 ：The function is a product of two functions, \(4x^2 - x\) and \(\ln(x)\). To find the derivative of this function, we can use the product rule, 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 2 ：So, we need to find the derivatives of \(4x^2 - x\) and \(\ln(x)\) first. The derivative of \(4x^2 - x\) is \(8x - 1\) and the derivative of \(\ln(x)\) is \(1/x\).

Step 3 ：Now that we have the derivatives of \(4x^2 - x\) and \(\ln(x)\), we can apply the product rule to find the derivative of the function.

Step 4 ：Applying the product rule, we get \((8x - 1)\ln(x) + (4x^2 - x)/x\).

Step 5 ：Simplifying the above expression, we get \((8x - 1)\ln(x) + 4x - 1\).

Step 6 ：Final Answer: The derivative of the function \(f(x)=(4x^{2}-x)\ln(x)\) is \(\boxed{(8x - 1)\ln(x) + 4x - 1}\)