Prove that the derivative of an odd function is even

[NWeid1]

A function f is an even function is f(-x)=f(x) for all x and is an odd function is f(-x)=-f(x) for all x. Prove that the derivative of an odd function is even and the derivative of an even function is off. I get what even and odd functions are but I'm not sure how to rigorously prove this. Thanks.

 

[Mark44]

A function f is an even function is f(-x)=f(x) for all x and is an odd function is f(-x)=-f(x) for all x. Prove that the derivative of an odd function is even and the derivative of an even function is off. I get what even and odd functions are but I'm not sure how to rigorously prove this. Thanks.

One way to do this is to use the limit definition of the derivative.

PS - don't delete the parts of the posting template.

 

[HallsofIvy]

Or, and I think simpler, use the chain rule to differentiate f(-x) with respect to x.