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What is the actual definition of Left Hand Derivative?

I bumped into this site and the second white box on their site gives the definition. Is that wrong?
What is the correct one then?

Klosew
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  • Left hand Derivative is just left derivative. Instead like taking derivative from both sides of the def of derivative, left derivative only take the limit from left side. – Brian Ding Feb 21 '15 at 06:16
  • @BrianDing Can you please check that link ? It says something else though I totally agree with you. – Klosew Feb 21 '15 at 06:17
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    I check the link actually and said the above. A second look shows that the def using $\lim_{h\rightarrow a-} \frac{f(a+h)-f(a)}{h}$ is problematic and it should be $\lim_{h\rightarrow 0-} \frac{f(a+h)-f(a)}{h}$ – Brian Ding Feb 21 '15 at 06:22

1 Answers1

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The left-hand and right-hand derivatives of $f$ at $a$ are defined by $$ f'_{-}(a)=\lim_{h\to 0^-}\frac{f(a+h)-f(a)}{h} $$ and $$ f'_{+}(a)=\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h} $$ if these limits exist. Then $f'(a)$ exists if and only if these one-sided derivatives exist and are equal.

Daniel W. Farlow
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  • Finally right... – Brian Ding Feb 21 '15 at 06:33
  • Is it correct to write the right-hand derivative as $$ f'{-}(a)=\lim{h\to 0^+}\frac{f(a-h)-f(a)}{-h} $$

    (basically replacing $ lim_{h\to 0^-}$ by $lim_{-h\to 0^+}$)?

     – User Mar 14 '18 at 02:42
  • @InternetGuy yes. See https://math.stackexchange.com/a/232681/587094 – Doesbaddel Feb 10 '19 at 09:50
  • @DanielWFarlow don't f'_ (a) and $f'(a^+)$ mean the same thing? – harry Nov 17 '20 at 16:47
  • @User There are two different secant functions of $f$ at $a$ that one can use: $S_1(x)=\frac{f(a)-f(a-x)}{a-(a-x)}$ and $S_2(x)=\frac{f(a+x)-f(a)}{(a+x)-a}$. It should be straightforward to see that $S_1(x)=S_2(-x)$, from which one can show (using $\delta-\varepsilon$ arguments) that each type of one-handed derivative has two valid definitions:

    $\displaystyle f'-(a):=\lim{x\to 0^+}\frac{f(a)-f(a-x)}{a-(a-x)} \iff \lim_{x \to 0^-}\frac{f(a+x)-f(a)}{(a+x)-a}$

    $\displaystyle f'+(a):=\lim{x\to0^-}\frac{f(a)-f(a-x)}{a-(a-x)} \iff \lim_{x \to 0^+}\frac{f(a+x)-f(a)}{(a+x)-a}$

     – S.C. Apr 07 '25 at 17:54