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