Odd Functions and Even Functions
Definitions
- A function $f(x)$ that satisfies $f(-x) = f(x)$ is called an Even function.
- A function $f(x)$ that satisfies $f(-x) = -f(x)$ is called an Odd function.
Description
Even functions are symmetric about the $y$ axis in the coordinate plane, while Odd functions are symmetric about the origin $O$.
For example, among the trigonometric functions, $\sin$ is Odd and $\cos$ is Even. Differentiating $\sin$ yields $\cos$, and differentiating $\cos$ yields $\sin$. It might seem unnecessary, but it’s useful in situations where you need not know the function exactly.
Derivatives
If $f$ is differentiable over all real numbers, the following holds:
- [1] The derivative of an Even function is an Odd function.
- [2] The derivative of an Odd function is an Even function.
Derivation
Let $f(x)$ be any Odd function, and $g(x)$ be any Even function.
Because of $f(x)=-f(-x)$, we have $$ f ' (x)=f ' (-x) $$ Because of $g(x)=g(-x)$, we have $$ g ' (x)=-g ' (-x) $$
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Corollary
Another good thing to know is that the derivative of an Even function $g(x)$, $g ' (x)$, is always $g ' (0)=0$.
Proof
$$ \begin{align*} & g ' (x)=-g ' (-x) \\ \implies& g ' (0)=-g ' (-0) \\ \implies& 2g ' (0)=0 \\ \implies& g ' (0)=0 \end{align*} $$
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