Question

find the derivative of y with respect to x.
y = 3\ln(\cos x)
\frac{dy}{dx}=\square

Explanation:

Step1: Apply chain - rule

Let $u = \cos x$, then $y = 3\ln(u)$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$.
The derivative of $y = 3\ln(u)$ with respect to $u$ is $\frac{dy}{du}=\frac{3}{u}$.

Step2: Find $\frac{du}{dx}$

The derivative of $u=\cos x$ with respect to $x$ is $\frac{du}{dx}=-\sin x$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $\frac{dy}{du}$ and $\frac{du}{dx}$ into the chain - rule formula: $\frac{dy}{dx}=\frac{3}{u}\cdot(-\sin x)$. Since $u = \cos x$, we have $\frac{dy}{dx}=\frac{3}{\cos x}\cdot(-\sin x)=- 3\tan x$.

Answer:

$-3\tan x$