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The function is ( y = \ln(\cos x) ).To find the derivative, use the chain rule: ( \frac{d}{dx}(\ln(u)) = \frac{1}{u} \frac{du}{dx} ).Here, ( u = \cos x ), so ( \frac{du}{dx} = \frac{d}{dx}(\cos x) = -\sin x ).Therefore, ( \frac{dy}{dx} = \frac{1}{\cos x} \cdot (-\sin x) )( \frac{dy}{dx} = -\frac{\sin x}{\cos x} )Since ( \frac{\sin x}{\cos x} = \tan x ), we have:( \frac{dy}{dx} = -\tan x ).This matches option C.