The derivative of $\log \{\log [\log (\sin \mathrm{x})\left]\right\}$ w.r.t. sin x is

$$\begin{gathered} Now \frac{d}{dx}(\log \left\{\log \left(\log \right(\sin x\left)\right)\right\} \\ =\frac{1}{\log \log \sin \mathrm{x}}\cdot \frac{d}{dx}(\log \left(\log \right(\sin x\left)\right)) \\ =\frac{1}{\log \log \sin \mathrm{x}}\frac{1}{\log \sin \mathrm{x}}\frac{d}{dx}(\log \left(\sin x\right)) \\ =\frac{1}{\log \log \sin \mathrm{x}}\cdot \frac{1}{\log \sin \mathrm{x}}\cdot \frac{1}{\sin \mathrm{x}}\cdot \cos \mathrm{x} \\ \mathrm{Now} \mathrm{required} \mathrm{derivative} \\ =\frac{\frac{1}{\log \log \sin \mathrm{x}}\cdot \frac{1}{\log \sin \mathrm{x}}\cdot \frac{1}{\sin \mathrm{x}}\cdot \cos \mathrm{x}}{\cos \mathrm{x}} \\ =\frac{1}{\sin \mathrm{x}\cdot \log \sin \mathrm{x}\cdot \log \log \sin \mathrm{x}} \end{gathered}$$