1. **State the problem:** Find the derivative $y'$ of the function $y = (\ln(x))^6$. 2. **Recall the formula:** To differentiate a composite function like $y = [u(x)]^n$, use the chain rule: $$y' = n [u(x)]^{n-1} \cdot u'(x)$$ where $u(x) = \ln(x)$ and $n = 6$. 3. **Differentiate the inner function:** The derivative of $u(x) = \ln(x)$ is $$u'(x) = \frac{1}{x}$$ 4. **Apply the chain rule:** $$y' = 6 (\ln(x))^{5} \cdot \frac{1}{x}$$ 5. **Final answer:** $$y' = \frac{6 (\ln(x))^{5}}{x}$$ This means the rate of change of $y$ with respect to $x$ depends on both the logarithm of $x$ raised to the fifth power and inversely on $x$ itself.