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Problem 63 Hard Difficulty

Recall that a function $ f $ is called \textit{even} if $ f(-x) = f(x) $ for all $ x $ in its domain and \textit{odd} if $ f(-x) = -f(x) $ for all such $ x $. Prove each of the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.

Answer

(a)$-\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}=-f^{\prime}(x)$ ; $f^{\prime}$ is odd
(b) $\lim _{\Delta x \rightarrow 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}=f^{\prime}(x)$ ; $f^{\prime}$ is even

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Video Transcript

This is problem number sixty three of this two hour calculus. Eighth edition, Section two, point eight. Recall that a function F is called even if half of negative X is equal to FX for all accidents domain and odd. If ever negative X is equal to negative epics for all such x prove each of the following party, the dura and even function is in our function. Part of being the dirt of oven on function is uneven function. So what we're going to do is we're gonna start with part A were to use our definition of a director, and we're going to assume, ah, whatever is given to us and to try to prove exactly that assumption. So, for example, the first part party, the derivative of an even function. So we're going to start with an even function where FX or the function has the property of when we played in negative X, you still get the same function out. The narrative of this even function is an odd function. So if the derivative is an odd function than it has to have the property that when you plug in at value, such as negative, eh an odd function should have, Ah, this equal to the derivative of at that value, the opposite of that. So negative there. So this is what we're trying to prove came. So we will use our definition here limit as except purchasing over the function after thanks minus f a p or we're playing and negativity. So I should where frame shins are. Yeah, can put his negative, eh? We're pushing. They're gonna be here. We're pushing it a bit here in the function and then already here, we're doing thanks. Minus negativity. Okay. The next day, we want to use the property of the function to function in part A is that the function is even. That's the assumption. So if the function is even on when we plug in a negative number, we get the same function as if we were to have plugged in a positive numbers. So this will be the same as f of eight here. But if the denominator will have X plus a king here, we're going to have to make a substitution. We're gonna let you equal negative X. So this is just a substitution, Charlie changing the variable from X to you. So if you is negative, x X is thinking of you game and this allows us to flip the sign of a here in the LTD. So here is exactly what we will see. The limit is you. We changed that from x two. You meaning that negative a flips and signs since he was negative. X So we went from ex to negative view and then negative. You flipped positive you That means negative a purports to positivity. Um and then we have X is thinking of you. So the function appears. Ah, function evaluated A negative view minus everyday and then And the denominator we have, ah, native X, which is negative. You plus a great Our next step is to again recall that our function at our function is even if our function is even when we and put a negative number it's the same as if we would have and put a positive number. So after negative use, the same was said of you minus f a b divided by and then here. We should notices we can factor out negative and called us. You might be saying the general tonight neither, and this is an important part here because it's negative. Can be factor out of the limit. So what is remaining? We have remaining Limited's your produce a half ofyou minus f obey over. You minus. If this looks familiar, it is. It is the definition of the derivative. So we have concluded here is that this is exactly equal to negative since the negative respected out negative f crime of eh? And this is exactly the conclusion we're getting to that because we assumed that it function was even to begin with, we did the derivative, um, and we plugged in a negative, eh? And what we ended the beginning was the opposite of the derivative function. Evaluated, eh? Which is exactly the definition of a non function. So we prove that the turret of uneven function is an odd function. And part B, we're going to do the same except switch. The assumptions we're going to soon to the original function is odd. And then we're going to attempt to prove that it is that this derivative waited at negativity, gives us the derivative at prime evaluated a positive it because that's the property of uneven function came. So we're going to do this limit as X approaches. A I want this function or we're approaching negative, eh? Correct this because we're plugging in negativity of this function f of X, which is a non function. That's a new assumption of part B minus f evaluated at negativity, directed by X minus. I need to be okay. Our next step when it is expert listening today, Um, our function here for Vic stays the same. But here for negative, eh? Well, we're taking the dirt of oven on function. F is not function in part B. If it's odd, then effort. Negative X is the same as the rest of X. So as a negative is the same as negative effort. Be so this negative in the front changes so positive ever, eh? You know, I have the negative aim there. The denominator is X plus eight. So we use at this point the same sir petition we had earlier. Same substitution. Where we let you equal to negative X and X. A negative. You on this serves to switch the sign here of the negative, eh? We get the limit as X approaches Are we substituted x with you. It flips the sign for the A So we're now approaching positivity. You purchase property of f evaluated X X. We substituted with negative year that goes there plus after pain guided by eggs plus a Okay. Oh, I always have to substitute this x here because X we replaced with negative, you two. So this was the replacement step. We went from X here, here and here to you. Um negative. You hear in here And then the negative, you hear just served to switch the sign of the A. So it was negative you reported a negativity. Now it's you purchase positivity on the Now we're going to use the property of the function. Your purchase A So this function f We are assuming in this herpes you that it is an odd function. If it's an odd function, then half of negative you will be equal to negative Half of you tell us today guided by negative and you place, eh? Here's the important step. Our next part will be to factor not just one negative one from the numerator, but also negative one from the denominator and the numerator will have negative one times after a half of you minus everything and in the denominator will have a negative one factored out, leaving you minus a and the denominator. So this negative here cancels out with is negative and what were left over with is the limited your purchase A or the quantity effort, humanise effort They over humanity and you can be any variable. It's a dummy variable. It could have been X. It's exactly the same as this definition of the derivative that we encountered initially that we use initially. So for you replaced X, it doesn't matter. It just serves This entire quantity is exactly equal to the derivative of F Evaluated adds a So what we did is we started with, uh, the derivative of effort at Negative Ain and with the assumption that the function after odd we were able to conclude it, that derivative function evaluated it positive is the same, instituted a functioning body with the negativity which is a property of even functions. So we approved, prevented party. Now that the derivative of an odd function is an even function on that concludes the proof for boats parts A and B