How would you prove that if is an even function ( ) and has a derivative at every point, then the derivative is an odd function ( ) ?
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Originally Posted by frenchguy87 How would you prove that if is an even function ( ) and has a derivative at every point, then the derivative is an odd function ( ) ? Apply the chain rule to , and thus ... Tonio
Originally Posted by tonio Apply the chain rule to , and thus ... Tonio How do you get to from that ?
Originally Posted by frenchguy87 How do you get to from that ? Well, since , right? Tonio
I don't know why but I'm not convinced
Originally Posted by frenchguy87 I don't know why but I'm not convinced Perhaps because you don't understand...? If then ; OTOH, applying the chain rule we have that and thus we get ... Tonio
Tonio gave the true answer. If you need some intuition, just draw an even function (symmetric around the y-axis) and pick some point x. Then draw tangent lines at x and at -x. Clearly they have opposite slopes.
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