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

Step 1 ：The given function is a product of two functions, \(x^6\) and \(\ln 5x\). To find the derivative of this function, we can use the product rule. The product rule 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 ：We need to find the derivatives of \(x^6\) and \(\ln 5x\) first. The derivative of \(x^6\) is \(6x^5\).

Step 3 ：The derivative of \(\ln 5x\) is a bit more complicated. We can use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function. The outer function is \(\ln x\) and the inner function is \(5x\). The derivative of \(\ln x\) is \(1/x\) and the derivative of \(5x\) is \(5\). So, the derivative of \(\ln 5x\) is \((1/x)*5 = 5/x\).

Step 4 ：Now we can apply the product rule. The derivative of the function \(f(x) = x^6 \ln 5x\) is \(f'(x) = 6x^5 \ln 5x + x^5\).

Step 5 ：Final Answer: \(f^{\prime}(x) = \boxed{6x^{5} \ln 5x + x^{5}}\)