\( \left. \begin{array} { l } { f ( x ) = ( \cos x ) ^ { x } } \\ { [ \ln ( \cos x ) - x \tan x ] ( \cos x ) ^ { x } } \\ { x ( \cos x ) ^ { x - 1 } ( \ln \cos x ) ( - \sin x ) } \\ { ( \cos x ) ^ { x } ( \ln \cos x ) ( - \sin x ) } \\ { x ( \cos x ) ^ { x - 1 } } \end{array} \right. \)

It looks like you're working with the function \( f(x) = (\cos x)^x \) and possibly trying to find its derivative or analyze its behavior. Let's go through the steps to differentiate this function. To differentiate \( f(x) = (\cos x)^x \), we can use logarithmic differentiation. Here are the steps: 1. **Take the natural logarithm of both sides**: \[ \ln f(x) = \ln((\cos x)^x) = x \ln(\cos x) \] 2. **Differentiate both sides with respect to \( x \)**: Using the product rule on the right side: \[ \frac{d}{dx}(\ln f(x)) = \frac{1}{f(x)} f'(x) \] \[ \frac{d}{dx}(x \ln(\cos x)) = \ln(\cos x) + x \frac{d}{dx}(\ln(\cos x)) \] The derivative of \( \ln(\cos x) \) is: \[ \frac{d}{dx}(\ln(\cos x)) = -\tan x \] Therefore: \[ \frac{d}{dx}(x \ln(\cos x)) = \ln(\cos x) - x \tan x \] 3. **Set the derivatives equal**: \[ \frac{1}{f(x)} f'(x) = \ln(\cos x) - x \tan x \] 4. **Solve for \( f'(x) \)**: \[ f'(x) = f(x) \left( \ln(\cos x) - x \tan x \right) \] Substituting back \( f(x) = (\cos x)^x \): \[ f'(x) = (\cos x)^x \left( \ln(\cos x) - x \tan x \right) \] So, the derivative of \( f(x) = (\cos x)^x \) is: \[ f'(x) = (\cos x)^x \left( \ln(\cos x) - x \tan x \right) \] If you have any specific questions or need further assistance with this function, feel free to ask!