How to pove that if f is even then f' is odd by using the definition of derivative

Here we will prove that if f is an even then its derivative is an odd function by using the definition of the derivative. So remember, the derivative of an even function is odd. On the other hand, the derivative of an odd function is even. This calculus tutorial will help you with your calculus 1, college calculus and AP calculus classes!

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[philipyao]

if you differentiate both sides of the definition of the even function you get
-f’(-x)=f’(x) (chain rule)
so f’(-x)=-f’(x)
so f’(x) is odd

[Ninja]

Thank you very much. I got this as a homework question but had no idea to use the definition. I tried using a geometric/graphical argument with tangent lines.

[physicsi-problems]

nice demonstartion. If a function is even its deerivative is odd and vcv

[shivamchouhan]

is opposite also true, I mean if f'(x) is even, is f(x) also odd

[prabhpreetbhandariokok]

Thanks, very nicely explained sir, Appreciated

[kobethebeefinmathworld]

And we can now explain why the zero function is always both odd and even as a function (using the definition of derivative).

[GirishManjunathMusic]

Given: f(x) is even
To prove: f'(x) is odd

Using the definition of a derivative:
f'(x) = lim x→h (f(x) – f(h))/(x – h)
f'(–x) = lim x→h (f(–x) – f(h))/(–x – h)

As f(x) is even:
f'(–x) = lim x→h (f(x) – f(h))/–(x + h)

Replacing dummy variable h with dummy variable k = –h:
f'(–x) = lim x→k (f(x) – f(–k))/–(x – k)

As f(x) is even:
f'(–x) = lim x→k (f(x) – f(k))/–(x – k)
= – lim x→k (f(x) – f(k))/(x – k)

Replacing dummy variable k with dummy variable h = k:
f'(–x) = – lim x→h (f(x) – f(h))/(x – h)
f'(–x) = –f'(x)

I'm not sure if this works entirely, the other definition might be better.

[zykren]

is this true for the opposite too? If f is odd then is f' even?

[mrminer]

Isn't this intuitively obvious, if you imagine the function written in power series?

[rofficial]

Thank you sir❤ Allah bless you with happiness ❤

[dextermcgiven]

Just differentiate both sides of what is given using the chain rule and you get the proof.

[alexdemoura]

a LOT of red pen and black pen boxes... 😁😁😆😆

[norn-sama]

It might not be a very difficult problem but I'm a bit proud that I was able to do it^^

[heliocentric]

Any antiderivative of an odd function is even.
Some antiderivatives of an even function are odd.

[yoyoezzijr]

f is even, so f(x) = f(-x). differentiating both sides, then f’(x) = -f’(-x), hence f’ is odd. proved.

[tetraktys]

Please, can you give us dificult equations, you makes me feel good at math So can you give us the solution of this equation? Just kidding, only I want the equation, please

[CGMossa]

Doesn't k approach 0 from the negative side now? Isn't that worthy a comment?