In a time series, the variations which occur due to general tendency of the data to increase or decrease over a long term are known as:

The derivative of $\log[\log(\log x^5)]$log[log(logx5)] with respect to $x$x requires the chain rule applied to nested functions.

For $y = \log(u)$y=log(u), the derivative is $\frac{dy}{dx} = \frac{1}{u} \cdot \frac{du}{dx}$dxdy​=u1​⋅dxdu​

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The outermost function is $\log[\log(\log x^5)]$log[log(logx5)].

$$\frac{d}{dx}\log[\log(\log x^5)] = \frac{1}{\log(\log x^5)} \times \frac{d}{dx}[\log(\log x^5)]$$dxd​log[log(logx5)]=log(logx5)1​×dxd​[log(logx5)]

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The middle function is $\log(\log x^5)$log(logx5).

$$\frac{d}{dx}[\log(\log x^5)] = \frac{1}{\log x^5} \times \frac{d}{dx}[\log x^5]$$dxd​[log(logx5)]=logx51​×dxd​[logx5]

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The innermost function is $\log x^5$logx5.

$$\frac{d}{dx}[\log x^5] = \frac{1}{x^5} \times \frac{d}{dx}[x^5]$$dxd​[logx5]=x51​×dxd​[x5]

$$\frac{d}{dx}[x^5] = 5x^4$$dxd​[x5]=5x4

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Combining all results:

$$\frac{d}{dx}\log[\log(\log x^5)] = \frac{1}{\log(\log x^5)} \times \frac{1}{\log x^5} \times \frac{1}{x^5} \times 5x^4$$dxd​log[log(logx5)]=log(logx5)1​×logx51​×x51​×5x4

$$= \frac{5x^4}{x^5 \cdot \log x^5 \cdot \log(\log x^5)}$$=x5⋅logx5⋅log(logx5)5x4​

$$= \frac{5}{x \cdot \log x^5 \cdot \log(\log x^5)}$$=x⋅logx5⋅log(logx5)5​

Therefore, the answer is $\dfrac{5}{x(\log x^5)\log(\log x^5)}$x(logx5)log(logx5)5​