Differentiate the following w.r.t. x: [ ( ^5) ] - Vidyadip

Question

Differentiate the following w.r.t. x:
$\log\big[\log(\log\text{x}^5)\big]$

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Answer

Let $\text{y}=\log\big[\log(\log\text{x}^5)\big]$
$\therefore\ \frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\log\big[\log(\log\text{x}^5)\big]$
$=\frac{1}{\log\log\text{x}^5}\cdot\frac{\text{d}}{\text{dx}}\big(\log\cdot\log\text{x}^5\big)$
$=\frac{1}{\log\log\text{x}^5}\cdot\frac{1}{\log\text{x}^5}\cdot\frac{\text{d}}{\text{dx}}\log\text{x}^5$
$=\frac{1}{\log\log\text{x}^5}\cdot\frac{1}{\log\text{x}^5}\cdot\frac{\text{d}}{\text{dx}}(5\log\text{x})$
$=\frac{5}{\text{x}\cdot\log(\log\text{x}^5)\cdot\log(\text{x}^5)}$

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