True or false?

Question

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

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Answer 1 · 2018-04-07T13:19:25

Shown below using the chain rule:

$d/dx f ( g(x) ) = g'(x) f'( g(x) )$

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

$=> d/dx f(-x) = d/dx f(x) = f'(x)$

We can also use the chain rule:

$=> d/dx f(-x) = -f'(-x)$

$therefore f'(x) = - f'(-x)$

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

Where this is the definition of the odd function

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

$=> d/dx g(-x) = d/dx -g(x) = -g'(x)$

Use chain rule: $d/dx g(-x) = -g'(-x)$

$=> -g'(-x) = -g'(x)$

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

This is the definition of the even function