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?
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.)