differentiate ( f(x)=\frac{9-cosh x}{5+cosh x} ). ( f^{prime}(x)= )

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Accepted Answer

# Explanation:
## Step1: Apply the quotient rule
The quotient rule is $(\frac{u}{v})^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Let $u = 9-\cosh x$, then $u^\prime=-\sinh x$. Let $v = 5+\cosh x$, then $v^\prime=\sinh x$.
## Step2: Substitute into the quotient rule formula
$$
\begin{align*}
f^\prime(x)&=\frac{(-\sinh x)(5 + \cosh x)-(9-\cosh x)(\sinh x)}{(5+\cosh x)^{2}}\
&=\frac{-5\sinh x-\sinh x\cosh x - 9\sinh x+\sinh x\cosh x}{(5+\cosh x)^{2}}\
&=\frac{-14\sinh x}{(5+\cosh x)^{2}}
\end{align*}
$$

# Answer:
$\frac{-14\sinh x}{(5 + \cosh x)^{2}}$

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Question

Explanation:

Step1: Apply the quotient rule

Step2: Substitute into the quotient rule formula

Answer: