Find the derivative of the function.
\[
f(x)=5 x \ln (x)
\]
\[
f^{\prime}(x)=
\]
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Step 1 ：The derivative of a function can be found using the rules of differentiation. In this case, we have a function that is the product of two simpler functions, 5x and ln(x). The product rule of differentiation 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. In this case, the first function is 5x and its derivative is 5. The second function is ln(x) and its derivative is 1/x.

Step 2 ：So, the derivative of the function \(f(x)=5x\ln(x)\) is \(f'(x)=5\ln(x) + 5\).

Step 3 ：\(\boxed{f'(x)=5\ln(x) + 5}\)