Derivative of Odd Function is Even

Theorem

Let $f$ be a differentiable real function such that $f$ is odd.

Then its derivative $f'$ is an even function.

Proof

\(\ds \map f x\)

\(=\)

\(\ds -\map f {-x}\)

Definition of Odd 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 even function.

$\blacksquare$