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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 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 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

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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 and 2 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