Limit Continuity And Differentiability Ques 108

Answer:

Correct Answer: 108.$(a, b, d)$

Solution:

108. Given, $y=\sec \left(\tan ^{-1} x\right)$

$$ \begin{alignedat} \text { Let } & & \tan ^{-1} x & =\theta \\ \Rightarrow & & x & =\tan \theta \\ \therefore & & y & =\sec \theta=\sqrt{1+x^{2}} \end{aligned} $$

On differentiating w.r.t. $x$, we get

$$ \begin{array}{rlrl} \frac{d y}{d x} & =\frac{1}{2 \sqrt{1+x^{2}}} \cdot 2 x \\ \text { At } x=1, & \frac{d y}{d x} & =\frac{1}{\sqrt{2}} \end{array} $$