Question Detail

The function is y = loge(cos x), which is equivalent to y = ln(cos x).To differentiate this, we use the chain rule: d/dx(ln(u)) = u'/u.Here, u = cos x.First, find the derivative of u with respect to x:u' = d/dx(cos x) = -sin xNow apply the chain rule:dy/dx = u'/u = (-sin x) / (cos x)Recall the trigonometric identity sin x / cos x = tan x.Therefore, dy/dx = -tan x.This matches option C.