1. The problem is to find the derivative of the function $f(x) = \ln(3x^5)$. 2. Recall the derivative rule for the natural logarithm: if $f(x) = \ln(g(x))$, then $f'(x) = \frac{g'(x)}{g(x)}$. 3. Here, $g(x) = 3x^5$. We need to find $g'(x)$. 4. Differentiate $g(x) = 3x^5$ using the power rule: $g'(x) = 3 \times 5x^{5-1} = 15x^4$. 5. Apply the derivative formula for the logarithm: $$f'(x) = \frac{g'(x)}{g(x)} = \frac{15x^4}{3x^5}.$$ 6. Simplify the fraction: $$f'(x) = \frac{\cancel{15}x^4}{\cancel{3}x^5} = 5 \times \frac{x^4}{x^5} = 5x^{4-5} = 5x^{-1} = \frac{5}{x}.$$ 7. The derivative is $f'(x) = \frac{5}{x}$. 8. Check the options: none of the options matches $\frac{5}{x}$. 9. Therefore, the correct answer is (e) None of the given options is correct.