1. The problem is to find the derivative $\frac{dy}{dx}$ of the function $f(x) = \ln(x^4)$.\n\n2. Recall the chain rule and the derivative of the natural logarithm function: if $f(x) = \ln(g(x))$, then $f'(x) = \frac{g'(x)}{g(x)}$.\n\n3. Here, $g(x) = x^4$. The derivative of $g(x)$ is $g'(x) = 4x^3$.\n\n4. Applying the chain rule, we get:\n$$\frac{dy}{dx} = \frac{4x^3}{x^4} = \frac{4x^3}{x^4} = \frac{4}{x}.$$\n\n5. Therefore, the derivative of $f(x) = \ln(x^4)$ is $\frac{4}{x}$.\n\n6. Among the options given, the correct answer is c. $\frac{4}{x}$.