# True or false?

## The derivative of an odd is an even function and the derivative of an even is an odd.

Apr 7, 2018

Shown below using the chain rule:

$\frac{d}{\mathrm{dx}} f \left(g \left(x\right)\right) = g ' \left(x\right) f ' \left(g \left(x\right)\right)$

#### Explanation:

An even function is know to be defined as: $f \left(- x\right) = f \left(x\right)$

$\implies \frac{d}{\mathrm{dx}} f \left(- x\right) = \frac{d}{\mathrm{dx}} f \left(x\right) = f ' \left(x\right)$

We can also use the chain rule:

$\implies \frac{d}{\mathrm{dx}} f \left(- x\right) = - f ' \left(- x\right)$

$\therefore f ' \left(x\right) = - f ' \left(- x\right)$

color(red)(therefore - f'(x) = f'(-x) " hence odd "

Where this is the definition of the odd function

The odd function is: $g \left(- x\right) = - g \left(x\right)$

$\implies \frac{d}{\mathrm{dx}} g \left(- x\right) = \frac{d}{\mathrm{dx}} - g \left(x\right) = - g ' \left(x\right)$

Use chain rule: $\frac{d}{\mathrm{dx}} g \left(- x\right) = - g ' \left(- x\right)$

$\implies - g ' \left(- x\right) = - g ' \left(x\right)$

color(red)(therefore g'(-x) = g'(x) " hence even"

This is the definition of the even function