Differentiate. \[ f(x)=x^{6} \ln 5 x \]

Step 1 ：The given function is a product of two functions, \(x^6\) and \(\ln(5x)\).

Step 2 ：To differentiate this function, we will use the product rule of differentiation which states that the derivative of a 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 ：The derivative of \(x^6\) is \(6x^5\).

Step 4 ：The derivative of \(\ln(5x)\) is \(\frac{1}{x}\).

Step 5 ：So, the derivative of the given function is \(6x^5 \ln(5x) + x^6 \times \frac{1}{x}\).

Step 6 ：Simplifying the expression, we get \(6x^5 \ln(5x) + x^5\).

Step 7 ：Final Answer: The derivative of the function \(f(x)=x^{6} \ln 5 x\) is \(\boxed{6x^{5} \ln(5x) + x^{5}}\).