If f(x) is an odd function, then write whether f'(x) is even of o… - Vidyadip

Question

If f(x) is an odd function, then write whether f'(x) is even of odd.

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Answer

We have, f(x) is an odd function.
$\Rightarrow\text{f}(-\text{x})=-\text{f}(\text{x})$
$\Rightarrow\frac{\text{d}}{\text{dx}}\big\{\text{f}(-\text{x})\big\}=-\frac{\text{d}}{\text{dx}}\big\{\text{f}(\text{x})\big\}$
$\Rightarrow\text{f}'(-\text{x})\frac{\text{d}}{\text{dx}}(-\text{x})=-\text{f}'\text{(x)}$
$\Rightarrow\text{f}'(-\text{x})\times(-1)=-\text{f}'\text{(x)}$
$\Rightarrow\text{f}'(-\text{x})=-\text{f}'\text{(x)}$
$\Rightarrow\text{f}'(-\text{x})=\text{f}'\text{(x)}$
Thus, f'(x) is an even function.

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