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The domain of the derivative of the function $f(x) = \left\{ \begin{array}{l}{\tan ^{ - 1}}x\;\;\;\;\;,\;|x|\; \le 1\\\frac{1}{2}(|x|\; - 1)\;,\;|x|\; > 1\end{array} \right.$ is
$R - \{ 0\} $
$R - \{ 1\} $
$R - \{ - 1\} $
$R - \{ - 1,\;1\} $
Solution
(c) $f(x) = \left\{ \begin{array}{l}\frac{1}{2}( – x – 1),\,\,\,x < – 1\\{\tan ^{ – 1}}x,\,\,\,\, – 1 \le x \le 1\\\frac{1}{2}(x + 1),\,\,\,x > 1\end{array} \right.$; $f'(x) = \left\{ \begin{array}{l} – \frac{1}{2},\,\,\,\,\,\,\,\,x < – 1\\\frac{1}{{1 + {x^2}}},\,\, – 1 < x < 1\\\frac{1}{2},\,\,\,\,\,\,\,\,\,\,\,\,x > 1\end{array} \right.$
$f'( – 1 – 0) = – \frac{1}{2};\,\,f'( – 1 + 0) = \frac{1}{{1 + {{( – 1 + 0)}^2}}} = \frac{1}{2}$
$f'(1 – 0) = \frac{1}{{1 + {{(1 – 0)}^2}}} = \frac{1}{2};\,\,f'(1 + 0) = \frac{1}{2}$
$\therefore$ $f'( – 1)$ does not exist;
$\therefore$ domain of $f'(x) = R – \{ – 1\} $.