The derivative of an even function is an odd function. The derivative of an odd function is an even function. Prove the results by from the limit definition of the derivative: Lim( as x approaches 0) [f(x - f(a)] / (x - a )

Question

The derivative of an even function is an odd function.  The derivative of an odd function is an even function.   Prove the results by from the limit definition of the derivative:  Lim( as x approaches 0)  [f(x - f(a)]  / (x - a )

anonymous · Answer

this is trivial if you get to use the chain rule.  but you must use definition yes?

anonymous · Answer

f(x)'=lim f(x)-f(a)/(x-a)
if it is an even function than we know that f(x)=f(-x)
f(-x)'=lim f(x)-f(a)/(-x-a)=-f(x)'

anonymous · Answer

same way if it is odd

anonymous · Answer

so we assume f is odd and prove f' is even in other word we have to show if 
$$f(-a)=-f(a)$$ then $$f'(-a)=f'(a)$$

anonymous · Answer

oh what andras said!