Even and Odd Functions
Definitions
- A function $f(x)$ satisfying $f(-x) = f(x)$ is called an even function.
- A function $f(x)$ satisfying $f(-x) = -f(x)$ is called an odd function.
Explanation
An even function is a function symmetric about the $y$-axis in the coordinate plane, and an odd function is a function symmetric about the origin $O$.
As examples, among the trigonometric functions we can point to $\sin$, which is odd, and $\cos$, which is even. Differentiating $\sin$ gives $\cos$, and differentiating $\cos$ gives $\sin$. It may not seem particularly useful, but it comes in handy in situations where you don’t need to know the function exactly.
Derivative
If $f$ is differentiable on the whole real line, the following hold.
- [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 an arbitrary odd function and $g(x)$ an arbitrary even function.
Since $f(x)=-f(-x)$, $$ f ' (x)=f ' (-x) $$ Since $g(x)=g(-x)$, $$ g ' (x)=-g ' (-x) $$
■
Corollary
One more thing worth knowing: the derivative $g ' (x)$ of an even function $g(x)$ always satisfies $g ' (0)=0$.
Proof
$$ \begin{align*} & g ' (x)=-g ' (-x) \\ \implies& g ' (0)=-g ' (-0) \\ \implies& 2g ' (0)=0 \\ \implies& g ' (0)=0 \end{align*} $$
■