1. The problem is to find the derivative of the function $f(x) = 5 \ln(4x)$.\n\n2. To differentiate $5 \ln(4x)$, yes, you need to use the chain rule because the argument of the logarithm is a function of $x$, specifically $4x$.\n\n3. The chain rule states that if you have a composite function $h(x) = g(f(x))$, then its derivative is $h'(x) = g'(f(x)) \cdot f'(x)$.\n\n4. Here, $g(u) = 5 \ln(u)$ and $f(x) = 4x$. So, $g'(u) = \frac{5}{u}$ and $f'(x) = 4$.\n\n5. Applying the chain rule, the derivative is:\n$$\frac{d}{dx} 5 \ln(4x) = 5 \cdot \frac{1}{4x} \cdot 4$$\n\n6. Simplify the expression by canceling the 4 in numerator and denominator:\n$$5 \cdot \frac{\cancel{4}}{4x} \cdot \cancel{4} = \frac{5}{x}$$\n\n7. So, the derivative is $\frac{5}{x}$.