Question:
If $f(x)$ is an even function, then write whether $f^{\prime}(x)$ is even or odd.
Solution:
$f(x)$ is an even function.
This means that $\mathrm{f}(-\mathrm{x})=\mathrm{f}(\mathrm{x})$.
If we differentiate this equation on both sides w.r.t. $x$, we get -
$f^{\prime}(-x) \cdot(-1)=f^{\prime}(x)$
or, $-f^{\prime}(-x)=f^{\prime}(x)$
i.e., $f^{\prime}(x)$ is an odd function. (Ans)
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