Question

The differential coefficient of $a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}$ is

• A. $\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{x\sqrt{x^2-1}}{\log }_{10}$ a
• B. $-\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{|x|\sqrt{x^2-1}}{\log }_{10}$ a
• C. $\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{|x|\sqrt{x^2-1}}{\log }_{a}10$
• D. $\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{x\sqrt{x^2-1}}{\log }_{a}10$

Answer 1

The correct option is B $-\frac{a^{{\log }_{10}\left({\text{cosec}}^{-1}x\right)}}{[\text{cosec}{]}^{-1}x}\frac{1}{|x|\sqrt{x^2-1}}{\log }_{10}$ a

$y=a^{{\log }_{10}\left({\csc }^{-1}x\right)}$
Differentiate both sides with respect to x
$\frac{dy}{dx}=a^{{\log }_{10}\left({\csc }^{-1}x\right)}{\log }_{10}a\frac{d}{dx}\left[{\log }_{10}{\csc }^{-1}x\right][\because \frac{d}{dx}{a}^x=a^x\log a]$
$\frac{dy}{dx}=a^{{\log }_{10}\left({\csc }^{-1}x\right)}\cdot {\log }_{10}a\cdot \frac{1}{{\csc }^{-1}x}\cdot \frac{d}{dx}\left({\csc }^{-1}x\right)$
$\frac{dy}{dx}=a^{{\log }_{10}\left({\csc }^{-1}x\right)}\cdot {\log }_{10}a\cdot \frac{1}{{\csc }^{-1}x}\cdot \left[\frac{-1}{\left|x\right|\sqrt{x^2-1}}\right]\cdot 1$
$\frac{dy}{dx}=\frac{-a^{{\log }_{10}\left({\csc }^{-1}x\right)}}{\left({\csc }^{-1}x\right)}\frac{1}{\left|x\right|\sqrt{x^2-1}}{\log }_{10}a$