Find the derivative of the function. \[ \begin{array}{r} f(x)=5 x \ln (x) \\ f^{\prime}(x)=\square \end{array} \]

Step 1 ：Given the function \(f(x)=5x\ln(x)\).

Step 2 ：We need to find the derivative of this function.

Step 3 ：We can use the product rule to find the derivative. The product rule states that the derivative of two functions multiplied together is the first function times the derivative of the second function plus the second function times the derivative of the first function.

Step 4 ：In this case, our first function is \(5x\) and our second function is \(\ln(x)\).

Step 5 ：The derivative of \(5x\) is \(5\) and the derivative of \(\ln(x)\) is \(\frac{1}{x}\).

Step 6 ：Applying the product rule, we get \(f'(x)=5x\cdot\frac{1}{x} + 5\ln(x)\).

Step 7 ：Simplifying this, we get \(f'(x)=5\ln(x) + 5\).

Step 8 ：So, the derivative of the function \(f(x)=5x\ln(x)\) is \(f'(x)=5\ln(x) + 5\).

Step 9 ：\(\boxed{f'(x)=5\ln(x) + 5}\) is the final answer.