Odd Functions and Their Derivatives: A Theorem

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SUMMARY

The derivative of an odd function, defined by the property f(-x) = -f(x), is an even function, satisfying f'(-x) = f'(x). This conclusion is derived using the chain rule, where (f(-x))' = -f'(-x) leads to the result that f' is even. The discussion confirms the theorem that connects the properties of odd and even functions through their derivatives, establishing a clear mathematical relationship.

PREREQUISITES
  • Understanding of odd and even functions in mathematics
  • Familiarity with calculus, specifically derivatives
  • Knowledge of the chain rule in differentiation
  • Basic mathematical notation and terminology
NEXT STEPS
  • Study the properties of even and odd functions in greater detail
  • Explore the implications of the chain rule in calculus
  • Investigate other theorems related to function derivatives
  • Practice problems involving odd and even functions and their derivatives
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Mathematics students, educators, and anyone interested in calculus and function properties will benefit from this discussion.

Zaare
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It seems the derivate of an odd function [tex](f(-x)=-f(x))[/tex] is an even function [tex](f(-x)=f(x))[/tex], and vice versa. Is there a theroem about this?
 
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Suppose f is odd. We have that (f(-x))' = (-f(x))' = -f'(x). But by the chain rule, (f(-x))' = -f'(-x). Thus -f'(-x) = -f'(x) <=> f'(-x) = f'(x) <=> f' is even.
 
Ah, that was easy. Thank you. :)
 

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