1. **State the problem:** Find the derivative of the function $$y = \ln(\cos(x))^x$$ with respect to $$x$$. 2. **Rewrite the function:** The function can be written as $$y = (\ln(\cos(x)))^x$$. 3. **Use logarithmic differentiation:** To differentiate $$y = u^x$$ where $$u = \ln(\cos(x))$$, take the natural logarithm of both sides: $$\ln y = x \ln u$$ 4. **Differentiate both sides with respect to $$x$$:** $$\frac{1}{y} \frac{dy}{dx} = \ln u + x \frac{1}{u} \frac{du}{dx}$$ 5. **Find $$\frac{du}{dx}$$:** $$u = \ln(\cos(x))$$ $$\frac{du}{dx} = \frac{1}{\cos(x)} \cdot (-\sin(x)) = -\tan(x)$$ 6. **Substitute $$u$$ and $$\frac{du}{dx}$$ back:** $$\frac{1}{y} \frac{dy}{dx} = \ln(\ln(\cos(x))) + x \frac{1}{\ln(\cos(x))} (-\tan(x))$$ 7. **Multiply both sides by $$y$$ to isolate $$\frac{dy}{dx}$$:** $$\frac{dy}{dx} = (\ln(\cos(x)))^x \left[ \ln(\ln(\cos(x))) - \frac{x \tan(x)}{\ln(\cos(x))} \right]$$ 8. **Final answer:** $$\boxed{\frac{dy}{dx} = (\ln(\cos(x)))^x \left( \ln(\ln(\cos(x))) - \frac{x \tan(x)}{\ln(\cos(x))} \right)}$$