Question

If $y=[\log \log \sin x^{\circ }{]}^7$, find $\frac{dy}{dx}$

• A. $\frac{6\pi }{{180}^{\circ }}\left[\log \right(\log \sin x^{\circ }){]}^7.\frac{\tan x^{\circ }}{\log \sin x^{\circ }}$
• B. $\frac{7\pi }{{180}^{\circ }}\left[\log \right(\log \sin x^{\circ }){]}^6.\frac{\cot x^{\circ }}{\log \sin x^{\circ }}$
• C. $\frac{\pi }{{30}^{\circ }}\left[\log \right(\log \sin x^{\circ }){]}^7.\frac{\cot x^{\circ }}{\log \sin x^{\circ }}$
• D. none of these

Answer 1

The correct option is B $\frac{7\pi }{{180}^{\circ }}\left[\log \right(\log \sin x^{\circ }){]}^6.\frac{\cot x^{\circ }}{\log \sin x^{\circ }}$

Change $x^{\circ }\text{to}\frac{\pi x}{180}$ radians.

$y={\left[\log \left\{\log \sin \frac{\pi x}{180}\right\}\right]}^7$

$\therefore \frac{dy}{dx}=7{\left[\log \log \sin \frac{\pi x}{180}\right]}^6\frac{1}{\log \sin \left(\pi x/180\right)}\cdot \frac{1}{\sin \left(\frac{\pi x}{180}\right)}\times \cos \frac{\pi x}{180}\cdot \frac{\pi }{180}$

$=\frac{7\pi }{{180}^{\circ }}\left[\log \right(\log \sin x^{\circ }){]}^6.\frac{\cot x^{\circ }}{\log \sin x^{\circ }}$