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?
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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
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@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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2I 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
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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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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
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