find the derivative of y with respect to x.\ny = ln(cos x)\n\\frac{dy}{dx}=\\square

Explanation:

Step1: Apply the chain rule

The chain rule states that if (y = f(g(x))), then (y^\prime=f^\prime(g(x))\cdot g^\prime(x)). Here, (y = \ln(u)) where (u=\cos x). The derivative of (\ln(u)) with respect to (u) is (\frac{1}{u}), and the derivative of (\cos x) with respect to (x) is (-\sin x).

Step2: Substitute (u = \cos x)

[ \begin{align*} \frac{dy}{dx}&=\frac{1}{\cos x}\cdot(-\sin x)\ &=-\frac{\sin x}{\cos x} \end{align*} ]

Step3: Simplify the expression

Since (\frac{\sin x}{\cos x}=\tan x), then (-\frac{\sin x}{\cos x}=-\tan x)

Answer:

(-\tan x)