Question

find the derivative of y with respect to x.
y = 4 ln ( sin x)
\frac{dy}{dx}=\square

Explanation:

Step1: Apply chain - rule

The chain - rule states that if $y = f(g(x))$, then $\frac{dy}{dx}=f^\prime(g(x))\cdot g^\prime(x)$. Here, $y = 4\ln(u)$ where $u=\sin x$. First, find the derivative of $y$ with respect to $u$ and $u$ with respect to $x$. The derivative of $\ln(u)$ with respect to $u$ is $\frac{1}{u}$, so the derivative of $4\ln(u)$ with respect to $u$ is $\frac{4}{u}$.

Step2: Find derivative of inner function

The derivative of $u = \sin x$ with respect to $x$ is $\cos x$.

Step3: Multiply the two derivatives

By the chain - rule $\frac{dy}{dx}=\frac{4}{u}\cdot\cos x$. Substitute $u = \sin x$ back into the equation.
$\frac{dy}{dx}=\frac{4\cos x}{\sin x}$.
Since $\frac{\cos x}{\sin x}=\cot x$, we have $\frac{dy}{dx}=4\cot x$.

Answer:

$4\cot x$