[ ( x^5 ) ] का x के सापेक्ष अवकलन है-

MCQ

$\log \left[\log \left(\log x^5\right)\right]$ का $x$ के सापेक्ष अवकलन है-

✓
$\frac{5}{x \log \left(x^5\right) \log \left(\log x^5\right)}$
B
$\frac{5}{x \log \left(\log x^5\right)}$
C
$\frac{5 x^4}{\log \left(x^5\right) \log \left(\log x^5\right)}$
D
$\frac{5 x^4}{\log x^5 \log \left(\log x^5\right)}$

✓

Answer

Correct option: A.

$\frac{5}{x \log \left(x^5\right) \log \left(\log x^5\right)}$

(A) $\frac{5}{x \log \left(x^5\right) \log \left(\log x^5\right)}$
माना $y=\log \left[\log \left(\log x^5\right)\right]$
$
\begin{aligned}
\therefore \quad \frac{d y}{d x} & =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{d}{d x}\left[\log \left(\log x^5\right)\right] \\
& =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log x^5} \frac{d}{d x}\left(\log x^5\right) \\
& =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log x^5} \cdot \frac{1}{x^5} \cdot \frac{d}{d x}\left(x^5\right) \\
& =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log x^5} \cdot \frac{1}{x^5} \cdot 5 x^4 \\
& =\frac{5}{x\left(\log x^5\right) \log \left(\log x^5\right)}
\end{aligned}
$

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