Question

If f (x) is an even function, then write whether `f' (x)` is even or odd ?

Answer 1

\[\text {We have }, f\left( x \right) \text{ is an even function} . \]

\[ \Rightarrow f\left( - x \right) = f\left( x \right)\]

\[\Rightarrow \frac{d}{dx}\left\{ f\left( - x \right) \right\} = \frac{d}{dx}\left\{ f\left( x \right) \right\}\]

\[ \Rightarrow f'\left( - x \right)\frac{d}{dx}\left( - x \right) = f'\left( x \right)\]

\[ \Rightarrow f'\left( - x \right) \times \left( - 1 \right) = f'\left( x \right)\]

\[ \Rightarrow - f'\left( - x \right) = f'\left( x \right)\]

\[ \Rightarrow f'\left( - x \right) = - f'\left( x \right)\]

\[\text{ Thus,} f'\left( x \right) \text{ is an odd function } .\]