2019
Problem - 4529
Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.
Because $f(x)$ is odd, therefore
$$f(-x)=-f(x)\implies f(x) = -f(-x)$$
Taking derivative on both sides gives
$$f'(x) =(-f(-x))' = -(-f'(-x))=f'(-x)$$
The last step uses chain rule.