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Question
If f (x) is an odd function, then write whether `f' (x)` is even or odd ?
Solution
\[\text{We have,} f\left( x \right)\text { is an odd function}. \]
\[ \Rightarrow f\left( - x \right) = - f\left( x \right)\]
\[\Rightarrow \frac{d}{dx}\left\{ f\left( - x \right) \right\} = - \frac{d}{dx}\left\{ f\left( x \right) \right\}\]
\[ \Rightarrow f'\left( - x \right)\frac{d}{dx}\left( - x \right) = - f'\left( x \right)\]
\[ \Rightarrow f'\left( - x \right) \times \left( - 1 \right) = - f'\left( x \right)\]
\[ \Rightarrow - f'\left( - x \right) = - f'\left( x \right)\]
\[ \Rightarrow f'\left( - x \right) = f'\left( x \right)\]
\[\text{ Thus,} f'\left( x \right) \text{ is an even function}. \]
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