Answer
$f'$ is even.
Work Step by Step
Step 1. Let $f(x)$ be an odd function, we have $f(-x)=-f(x)$.
Step 2. Taking derivative on both sides, we get LHS=$f'(-x)=f'(-x)(-x)'=-f'(-x)$ and RHS=$-f'(x)$.
Step 3. Since LHS=RHS, we have $f'(-x)=f'(x)$ which means that the derivative becomes an even function.
