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But I had a thought that I can't quite wrap my around, and I think it's just because I don't fully understand the notation. I almost made this mistake but caught myself because I realized it doesn't work; it's essentially a silly "proof" for why the derivative of an odd function is odd:

1. f(-x)=-f(x), the definition of an odd function

2. Taking the derivative of both sides (since two functions which are equal for all values of x should also have equivalent derivatives):

f'(-x)=-f'(x)

I know this is wrong (since that says the derivative is also odd). I'm also pretty certain that the derivative of -f(x) is -f'(x) (because of the constant rule). So that means that the derivative of f(-x) is NOT f'(-x). My question: why is the derivative of f(-x) not f'(-x)?