True or false?

Question

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

1218 views around the world

You can reuse this answer
Creative Commons License

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