Question

If $f\left(\right.x\left.\right)=7^{\left(log\right)_{x} 7}$ where $x\in \left(\right.0,\infty \left.\right)-\left\{\right.1\left.\right\}$ , then $\left|f^{'} \left(\right. 7 \left.\right)\right|=$ _____

Answer 1

$f(x)=7^{\log _{x} 7}$
$\Rightarrow f^{\prime}(x)=7^{\log _{x} 7} \log 7 \cdot \frac{d}{d x}\left(\log _{x} 7\right)$
$=\left(7^{\log _{x} 7} \log 7\right)\left(\log 7 \cdot \frac{-1}{(\log x} \cdot \frac{1}{x}\right)$
$\Rightarrow f^{\prime}(7)=-7^{\log _{7} 7} \log 7 \cdot \log 7 \cdot \frac{1}{(\log 7} \cdot \frac{1}{7}$
$=-7 \cdot \frac{1}{7}=-1$