A recent tweet from @AnalysisFact noted that the derivative of an even function is an odd function. There are many ways to explore and understand this fact, but here’s a simple algebraic approach that uses a neat little trick in representing even and odd functions.
Claim: The derivative of a [differentiable] even function is an odd function.
Proof: Suppose is an even, differentiable function. Consider the function
First we show that . Since is even, we know
Now let’s differentiate . We have
where the last step follows by the chain rule.
we see that the derivative of is an odd function, as desired.