NIMCET 2018 — Mathematics PYQ

**Concept:**  

${\log }_{a}x=\frac{\log x}{\log a}$  

$\frac{d}{dx}\left(\frac{1}{x}\right)=\frac{-1}{x^2}$

Calculation:

Let $y={\log }_{10}x$ and $z={\log }_{x}10$

We have to find the value of $\frac{dy}{dz}$

$yz=\frac{\log x}{\log 10}\cdot \frac{\log 10}{\log x}=1$

$\Rightarrow y=\frac{1}{z}$

$\Rightarrow \frac{dy}{dz}=-\frac{1}{z^2}=-y^2=-({\log }_{10}x{)}^2$

$=-\frac{(\log x{)}^2}{(\log 10{)}^2}$

Hence, option (1) is correct.