Limit Continuity And Differentiability Ques 20

Answer:

Correct Answer: 20.$(\frac{-2\left(x^2-x-1\right)}{\left(x^2+1\right)^2} \cdot \sin ^2\left(\frac{2 x-1}{x^2+1}\right))$

Solution: Given, $\quad y=f\left(\frac{2 x-1}{x^2+1}\right)$

and $f^{\prime}(x)=\sin ^2 x$

$\therefore \quad \frac{d y}{d x}=f^{\prime}\left(\frac{2 x-1}{x^2+1}\right) \cdot \frac{d}{d x}\left(\frac{2 x-1}{x^2+1}\right)$

$=\sin ^2\left(\frac{2 x-1}{x^2+1}\right) \cdot\left\{\frac{\left(x^2+1\right) \cdot 2-(2 x-1)(2 x)}{\left(x^2+1\right)^2}\right\}$

$ \begin{aligned} & =\sin ^2\left(\frac{2 x-1}{x^2+1}\right) \cdot \frac{-2 x^2+2 x+2}{\left(x^2+1\right)^2} \\ & = \frac{-2\left(x^2-x-1\right)}{\left(x^2+1\right)^2} \sin ^2\left(\frac{2 x-1}{x^2+1}\right) \end{aligned} $