1. **State the problem:** Find the derivative with respect to $x$ of the function $$f(x) = \frac{(\ln(x))^3}{x^3}.$$\n\n2. **Formula and rules:** We will use the quotient rule for derivatives, which states:\n$$\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$$\nwhere $u = (\ln(x))^3$ and $v = x^3$.\n\n3. **Find derivatives of numerator and denominator:**\n- Derivative of $u = (\ln(x))^3$ using the chain rule:\n$$\frac{du}{dx} = 3(\ln(x))^2 \cdot \frac{1}{x} = \frac{3(\ln(x))^2}{x}.$$\n- Derivative of $v = x^3$:\n$$\frac{dv}{dx} = 3x^2.$$\n\n4. **Apply the quotient rule:**\n$$\frac{d}{dx}\left[\frac{(\ln(x))^3}{x^3}\right] = \frac{x^3 \cdot \frac{3(\ln(x))^2}{x} - (\ln(x))^3 \cdot 3x^2}{(x^3)^2}.$$\n\n5. **Simplify numerator:**\n$$x^3 \cdot \frac{3(\ln(x))^2}{x} = 3x^{3-1}(\ln(x))^2 = 3x^2(\ln(x))^2,$$\nso numerator becomes:\n$$3x^2(\ln(x))^2 - 3x^2(\ln(x))^3 = 3x^2 \left((\ln(x))^2 - (\ln(x))^3\right).$$\n\n6. **Simplify denominator:**\n$$(x^3)^2 = x^{6}.$$\n\n7. **Write the derivative:**\n$$\frac{3x^2 \left((\ln(x))^2 - (\ln(x))^3\right)}{x^6} = 3 \frac{\cancel{x^2}}{x^{6}} \left((\ln(x))^2 - (\ln(x))^3\right) = 3x^{2-6} \left((\ln(x))^2 - (\ln(x))^3\right) = 3x^{-4} (\ln(x))^2 (1 - \ln(x)).$$\n\n8. **Final answer:**\n$$\boxed{\frac{d}{dx}\left[\frac{(\ln(x))^3}{x^3}\right] = \frac{3(\ln(x))^2 (1 - \ln(x))}{x^4}}.$$