Solve the following systems of equations


$\log \left[\log \left(\log x^{5}\right)\right]$


Let, $\quad y=\log \left[\log \left(\log x^{5}\right)\right]$

Differentiating both sides w.r.t. $x$

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

$=\frac{1}{\log \left(\log x^{5}\right)} \times \frac{d}{d x} \log \left(\log x^{5}\right)$

$=\frac{1}{\log \left(\log x^{5}\right)} \times \frac{1}{\log \left(x^{5}\right)} \times \frac{d}{d x} \log x^{5}$

$=\frac{1}{\log \left(\log x^{5}\right)} \cdot \frac{1}{\log \left(x^{5}\right)} \cdot \frac{1}{x^{5}} \cdot \frac{d}{d x} x^{5}$

$=\frac{1}{\log \left(\log x^{5}\right)} \cdot \frac{1}{\log \left(x^{5}\right)} \cdot \frac{1}{x^{5}} \cdot 5 x^{4}$

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

Thus, $\quad \frac{d y}{d x}=\frac{5}{x \log \left(x^{5}\right) \cdot \log \left(\log x^{5}\right)}$.

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