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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)