Question

find the derivative of y with respect to x.
y = ln(cos x)
\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 \(u = \cos x\)

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Step3: Simplify the expression

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

Answer:

\(-\tan x\)