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

MCQ

Differentiation of $\log \left[\log \left(\log x^5\right)\right]$ $\text{w.r.t. x}$ 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.

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

$\therefore \frac{d y}{d x} =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{d}{d x}\left[\log \left(\log x^5\right)\right]$
$ =\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log x^5} \frac{d}{d x}\left(\log x^5\right)$
$=\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log x^5} \cdot \frac{1}{x^5} \cdot \frac{d}{d x}\left(x^5\right)$
$=\frac{1}{\log \left(\log x^5\right)} \cdot \frac{1}{\log x^5} \cdot \frac{1}{x^5} \cdot 5 x^4$
$=\frac{5}{x\left(\log x^5\right) \log \left(\log x^5\right)}$

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