Question

If $ y = \log \{ \log ( \log \:x)^2 \} $ then $\frac{dy}{dx}$ is equal to :

• A. $ \frac{1}{\log (\log \: x) }$
• B. $ \frac{1}{x \: \log (\log \: x) }$
• C. $ \frac{1}{x \:\log \: x . \log (\log \: x) }$
• D. $ \frac{1}{2x \:\log \: 2x . \log (\log \: x) }$

Answer 1

The Correct Option is C

Let $y = \log\left(\log \left(\log x\right)^{2}\right)$ $\therefore \:\:\: \frac{dy}{dx}=\frac{d}{dx} \left[\log \left\{\log \left(\log x\right)^{2}\right\}\right]$ $ = \frac{1}{\log \left(\log x\right)^{2}} \frac{d}{dx }\left[\log \left(\log x\right)^{2}\right]$ $=\frac{ 1}{\log\left(\log x\right)^{2}} \times\frac{1}{\left(\log x\right)^{2}} \times\frac{d}{dx}\left[\left(\log x\right)^{2}\right]$ $ = \frac{1}{\log \left(\log x\right)^{2}} \times\frac{1}{\left(\log x\right)^{2}} \times\frac{2 \log x}{x}$ $ = \frac{2}{x \log x.\log \left(\log x\right)^{2}} $ $= \frac{2}{2x \log x \log \left(\log x\right)} = \frac{1}{x \log x \log \left(\log x\right)}$