Differentiation of [ ( x^5 ) ] w.r.t. x is : - Vidyadip

MCQ

Differentiation of $\log \left[\log \left(\log x^5\right)\right]$ w.r.t. $x$ is :

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

✓

Answer

Correct option: D.

$\frac{5 x^4}{\log x^5 \log \left(\log x^5\right)}$

(D)Suppose $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)
\end{aligned}
$
$
\begin{array}{l}
=\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{array}
$

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