Use logarithmic differentiation to evaluate $y^{\prime}$.
\[
y=(5 x)^{\ln 5 x}
\]

Step 1 ：First, we take the natural logarithm of both sides of the equation to simplify the differentiation process. We get \(\ln y = \ln 5x \cdot \ln (5x)\).

Step 2 ：Next, we differentiate both sides of the equation with respect to \(x\). Using the product rule on the right side, we get \(\frac{y'}{y} = \ln (5x) \cdot \frac{1}{x} + 1\).

Step 3 ：Then, we multiply both sides of the equation by \(y\) to isolate \(y'\). We get \(y' = y \cdot \left(\ln (5x) \cdot \frac{1}{x} + 1\right)\).

Step 4 ：Finally, we substitute \(y = (5x)^{\ln 5x}\) back into the equation to get \(y' = (5x)^{\ln 5x} \cdot \left(\ln (5x) \cdot \frac{1}{x} + 1\right)\).

Step 5 ：So, the derivative of \(y = (5x)^{\ln 5x}\) is \(y' = (5x)^{\ln 5x} \cdot \left(\ln (5x) \cdot \frac{1}{x} + 1\right)\).