Question

If $f(x)=7^{{\log }_{x}7}$, where $x\in R^{+}-\{1\}$, find $f^{\prime }(7)$

Answer 1

Answer: -1

$f(x)=7^{{\log }_{x}7}$
$\Rightarrow f^{\prime }(x)=7^{{\log }_{x}7}\log 7\cdot \frac{d}{dx}\left({\log }_{x}7\right)$
$=\left(7^{{\log }_{x}7}\log 7\right)\left(\log 7\cdot \frac{-1}{(\log x{)}^2}\cdot \frac{1}{x}\right)$
$\Rightarrow f^{\prime }(7)=-7^{\log 7^7}\log 7\cdot \log 7\cdot \frac{1}{(\log 7{)}^2}\cdot \frac{1}{7}$
$=-7\cdot \frac{1}{7}=-1$