Differentiation of $\log[\log(\log x^5)]$ with respect to $x$ is

The derivative of $\log[\log(\log x^5)]$ with respect to $x$ requires the chain rule applied to nested functions.

For $y = \log(u)$, the derivative is $\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}$

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The outermost function is $\log[\log(\log x^5)]$.

$\frac{d}{dx}\log[\log(\log x^5)] = \frac{1}{\log(\log x^5)} \times \frac{d}{dx}[\log(\log x^5)]$

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The middle function is $\log(\log x^5)$.

$\frac{d}{dx}[\log(\log x^5)] = \frac{1}{\log x^5} \times \frac{d}{dx}[\log x^5]$

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The innermost function is $\log x^5$.

$\frac{d}{dx}[\log x^5] = \frac{1}{x^5} \times \frac{d}{dx}[x^5]$

$\frac{d}{dx}[x^5] = 5x^4$

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Combining all results:

$\frac{d}{dx}\log[\log(\log x^5)] = \frac{1}{\log(\log x^5)} \times \frac{1}{\log x^5} \times \frac{1}{x^5} \times 5x^4$

$= \frac{5x^4}{x^5 \cdot \log x^5 \cdot \log(\log x^5)}$

$= \frac{5}{x \cdot \log x^5 \cdot \log(\log x^5)}$

Therefore, the answer is $\dfrac{5}{x(\log x^5)\log(\log x^5)}$