frac{d}{dx} log_{sqrt{x}} left(frac{1}{x}righ | vedclass

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(D) Let $f(x) = \log_{\sqrt{x}} \left(\frac{1}{x}\right)$.Using the base change formula $\log_a b = \frac{\log b}{\log a}$,we have:$f(x) = \frac{\log(

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(D) Let $f(x) = \log_{\sqrt{x}} \left(\frac{1}{x}ight)$.Using the base change formula $\log_a b = \frac{\log b}{\log a}$,we have:$f(x) = \frac{\log(1/x)}{\log(\sqrt{x})}$Since $\log(1/x) = -\log x$ and $\log(\sqrt{x}) = \log(x^{1/2}) = \frac{1}{2} \log x$,we substitute these into the expression:$f(x) = \frac{-\log x}{\frac{1}{2} \log x}$For $x > 0$ and $x 
eq 1$,we can cancel $\log x$:$f(x) = \frac{-1}{1/2} = -2$Since $f(x) = -2$ is a constant function,its derivative with respect to $x$ is:$f'(x) = \frac{d}{dx}(-2) = 0$.

$\frac{d}{dx} \log_{\sqrt{x}} \left(\frac{1}{x}\right)$ is equal to

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