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

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}})