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What is the difference between $f'(a^+)$, $f'_+(a)$, $\lim_{x\to a^{+}}f'(x)$, and $\lim_{x\to a^{+}}\frac{f(x)-f(a)}{x-a}$? I assume they're all the same, but I'm not sure; don't they all essentially mean "the slope of the graph of f(x) just after the point x=a"?

Edit: I could used the left-hand limit with the negative sign, I know; I just used this for convenience. Other than that, is there any difference between any of these?

harry
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    The $+$ or $-$ refers to the "side" of the limit; the $+$ means that you're approaching the limit point from the right hand side, while $-$ means you're approaching from the left hand side. This might be a useful link if you haven't seen one-sided limits before: https://tutorial.math.lamar.edu/classes/calci/onesidedlimits.aspx – scoopfaze Nov 17 '20 at 17:06
  • I know that! It could be either + or - that I used, I used the first just because. – harry Nov 17 '20 at 17:18
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    Depends on context, I guess. At an elementary level, they can be thought to notate the same thing, which is the right hand derivative of $f$ at $x=a$; however, the second notation in your question can also refer to the [Dini derivative](https://en.wikipedia.org/wiki/Dini_derivative) which is not the same thing. Also, they are uncommon notations imo, best to go with the fourth one which is explicit. – Prasun Biswas Nov 17 '20 at 17:22
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    I would identify all four as left-hand derivatives, with some hesitation about the second. –  Nov 17 '20 at 17:22
  • @YvesDaoust [This](https://math.stackexchange.com/questions/1158510/left-hand-derivative-definition) is where I found that one. Looks legit. – harry Nov 17 '20 at 17:47
  • @scoopfaze: The third one is different. (See my answer below.) – Hans Lundmark Nov 17 '20 at 18:29
  • @YvesDaoust: You mean right-hand. Anyway, the second one is standard notation (at least it's what I learned in calculus class long time ago, and it's used by author such as Zorich and Stromberg, for example). It's the third one that you should hesitate about... – Hans Lundmark Nov 18 '20 at 06:31
  • @HansLundmark: yes I mean right-hand. –  Nov 18 '20 at 07:44

1 Answers1

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The third one, $\lim_{x \to a^+} f'(x)$, is different! If that limit exists and $f$ is continuous at $a$, then it equals the right-hand derivative $f'_+(a)$, as can be shown using the mean value theorem for derivatives. But it may happen that it doesn't exist even if $f'_+(a)$ exists, as shown by the standard examples of differentiable but not continuously differentiable functions. At it may also exist if $f$ is discontinuous at $a$, say $f(x)=0$ for $x \le a$ and $f(x)=1$ for $x>a$, where $\lim_{x \to a^+} f'(x) = \lim_{x \to a^+} 0 = 0$ but $f'_+(a)$ doesn't exist.

The second and the fourth one denote and define, respectively, the right-hand derivate.

Whether the first one, $f'(a^+)$, is supposed to mean the third or the second/fourth alternative is unclear to me. Usually $g(a^+)$ means the right-hand limit of $g$, and with that interpretation, $f'(a^+)$ would mean the right-hand limit of $f'$, i.e., the third alternative. But some people might also use it to denote the right-hand derivative. (And in many cases that's the same thing, so it might not matter much, but as I wrote above, it's not quite equivalent.)

Hans Lundmark
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  • The second one seems like it could also be used for the positive part of $f'$, evaluated at $a$. – Mark S. Nov 17 '20 at 19:06
  • Okay, so we know the second and the fourth ones are the same. Then, considering the first three alone, if we define f'(x) =g(x) for simplicity, these sources; [1](https://en.m.wikipedia.org/wiki/One-sided_limit) and [2](https://oregonstate.edu/instruct/mth251/cq/Stage3/Lesson/leftRighta.html) and more, all define $\lim_{x \to a^+} g(x) $ to be the right hand limit. Since $f'_+(a)$ is the right hand derivative, as you said, $g_+(a)$ becomes the right hand limit. Doesn't that mean all four are the same? – harry Nov 18 '20 at 03:00
  • @MarkS.: Well, it could, in principle, but it's a standard notation for the right-hand derivative (see for example Zorich, *Mathematical Analysis I*, p. 262). – Hans Lundmark Nov 18 '20 at 06:25
  • @HarryHolmes: No, they need bot be the same, as shown by the **counterexamples** I gave. I don't understand what you are trying to say with your function $g$, but whatever you mean, you should try it out on those counterexamples to see where your argument breaks down. – Hans Lundmark Nov 18 '20 at 06:28
  • I just replaced the function f'(x) with g(x) for simplicity. Considering that, $f'_+(a)=g_+(a)$, and the pages I linked seem to say $\lim_{x \to a^+} g(x)$(what we first defined as $\lim_{x \to a^+} f'(x)$) is the same as the right-hand limit. – harry Nov 18 '20 at 06:30
  • @HarryHolmes: What is $g_+(a)$? I never said anything about that notation. Did you mean $g(a^+)$? – Hans Lundmark Nov 18 '20 at 06:32
  • @MarkS.: Another standard textbook which uses $f'_+$ is Stromberg, *An Introduction to Classical Real Analysis*, p. 171. – Hans Lundmark Nov 18 '20 at 06:34
  • Yes, I meant $g(a^+)$, aren't they the same? – harry Nov 18 '20 at 06:35
  • No, when you define $f'_+(a)$ you consider a difference quotient “based at $a$” and let $h\to 0^+$, there's just one (one-sided) limit involved. When you consider $g(a^+)$ you first define $g(x)$ for $x>a$ by considering a difference quotient “based at $x$” and letting $h \to 0$ (ordinary limit, not one-sided) in that difference quotient, and then you take a *second* (one-sided) limit by letting $x \to a^+$ in the function $g(x)$. The results of these two different procedures are often the same, but *not necessarily*, as the counterexamples show. – Hans Lundmark Nov 18 '20 at 06:39
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    @HarryHolmes: By the way, when you write $g_+$ on its own, it usually means the positive part of $g$, which is the reason for the first comment above (by Mark S.). So when you see the notation $f'_+$, you shouldn't think of it as a subscript ${}_+$ attached to the function $f'$, but rather as the combined symbol ${}'_+$ attached to the function $f$. – Hans Lundmark Nov 18 '20 at 07:51