Limit Continuity and Differentiability 7 Question 17
17. The domain of the derivative of the functions $\tan ^{-1} x, \quad$ if $|x| \leq 1$ $f(x)=\frac{1}{2}(|x|-1), \quad$ if $|x|>1$ is
(2002, 2M)
(a) $R-{0}$
(b) $R-{1}$
(c) $R-{-1}$
(d) $R-{-1,1}$
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Answer:
Correct Answer: 17. (c)
Solution:
- Given that, $\log (x+y)=2 x y$
$\therefore$ At $x=0, \Rightarrow \log (y)=0 \Rightarrow y=1$
$\therefore$ To find $\frac{d y}{d x}$ at $(0,1)$
On differentiating Eq. (i) w.r.t. $x$, we get
$$ \begin{aligned} & \frac{1}{x+y} \quad 1+\frac{d y}{d y}=2 x \frac{d y}{d x}+2 y \cdot 1 \\ & \Rightarrow \quad \frac{d y}{d x}=\frac{2 y(x+y)-1}{1-2(x+y) x} \\ & \Rightarrow \quad \frac{d y}{d x} _{(0,1)}=1 \end{aligned} $$
