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  4. If f(x)=7 log x 7, where x ∈ R +- 1 find f'(7)

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

Limits and Derivatives

Solution:

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