91Ó°ÊÓ

To start, we need to simplify the expression inside the logarithm as much as possible. In this case, that means isolating the \(x\) in the denominator. $$f(x)=\log _{2} \frac{8}{\sqrt{x+1}} = \log _{2} \frac{8\sqrt{x+1}}{x+1}$$

Next, we'll use the properties of logarithms to rewrite the expression in a simpler form. Recall the following properties of logarithms: 1. \(\log_a{(x * y)} = \log_a{x} + \log_a{y}\) 2. \(\log_a{(\frac{x}{y})} = \log_a{x} - \log_a{y}\) 3. \(\log_a{x^n} = n\log_a{x}\) Using these properties, we can rewrite the given function: $$f(x) = \log _{2} \frac{8\sqrt{x+1}}{x+1} = \log _{2}{8} + \log _{2}{\sqrt{x+1}} - \log_{2}{(x+1)}.$$ Applying property 3, we obtain: $$f(x) = 3\log _{2}{2} + \frac{1}{2} \log _{2}{(x+1)} - \log_{2}{(x+1)}.$$

To find the derivative of \(f(x)\), we need to differentiate each term with respect to \(x\). Recall that the derivative of a \(\log_{a}{x}\) is \(\frac{1}{x\ln{a}}\): $$f'(x) = \frac{d}{dx}(3\log _{2}{2}) + \frac{d}{dx}(\frac{1}{2} \log _{2}{(x+1)}) - \frac{d}{dx}(\log_{2}{(x+1)}).$$ Since \(\log_{2}{2}\) is a constant, its derivative is 0: $$f'(x) = 0 + \frac{d}{dx}(\frac{1}{2} \log _{2}{(x+1)}) - \frac{d}{dx}(\log_{2}{(x+1)}).$$ Now, we differentiate the other two terms using the chain rule: $$f'(x) = 0 + \frac{1}{2}\cdot \frac{1}{(x+1)\ln{2}}\cdot 1 - \frac{1}{(x+1)\ln{2}}\cdot 1$$ By simplifying the expression and combining terms, we obtain the derivative with respect to \(x\): $$f'(x) = -\frac{1}{2(x+1)\ln{2}}.$$