find the derivative of the following function.
y = \frac{\csc w}{1 + \csc w}
\frac{dy}{dw}=\square

Question

Accepted Answer

# Explanation:
## Step1: Recall quotient - rule
The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = \csc w$, $v=1 + \csc w$.
## Step2: Find $u^\prime$ and $v^\prime$
The derivative of $\csc w$ is $-\csc w\cot w$. So, $u^\prime=-\csc w\cot w$ and $v^\prime=-\csc w\cot w$.
## Step3: Apply the quotient - rule
$$
\begin{align*}
\frac{dy}{dw}&=\frac{(-\csc w\cot w)(1 + \csc w)-\csc w(-\csc w\cot w)}{(1 + \csc w)^{2}}\
&=\frac{-\csc w\cot w-\csc^{2}w\cot w+\csc^{2}w\cot w}{(1 + \csc w)^{2}}\
&=\frac{-\csc w\cot w}{(1 + \csc w)^{2}}
\end{align*}
$$
# Answer:
$\frac{-\csc w\cot w}{(1 + \csc w)^{2}}$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall quotient - rule

Step2: Find $u^\prime$ and $v^\prime$

Step3: Apply the quotient - rule

Answer: