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

.

Thus, .

Now let’s differentiate .  We have

where the last step follows by the chain rule.

And since

we see that the derivative of is an odd function, as desired.

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