Derivative of Even Function is Odd
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Theorem
Let $f$ be a differentiable real function such that $f$ is even.
Then its derivative $f'$ is an odd function.
Proof
\(\ds \map f x\) | \(=\) | \(\ds \map f {-x}\) | Definition of Even Function | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac \d {\d x} \map f x\) | \(=\) | \(\ds \frac \d {\d x} \map f {-x}\) | differentiating both sides with respect to $x$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map {f'} x\) | \(=\) | \(\ds \map {f'} {-x} \times \paren {-1}\) | Chain Rule for Derivatives | ||||||||||
\(\ds \) | \(=\) | \(\ds -\map {f'} {-x}\) |
Hence the result by definition of odd function.
$\blacksquare$