We have an answer to "How are derivatives and one-sided derivatives related?"

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Calculus and Analysis
Differential Calculus
Derivatives

Kyran Hudson

Answered question

2021-02-12

How are derivatives and one-sided derivatives related?

Answer & Explanation

Delorenzoz

Skilled2021-02-13Added 91 answers

Step 1
One-sided derivatives are the derivatives in which ordinary limit is replaced by a one-sided limit.
Step 2
For a function f, one-sided derivatives at some point x = a are
$lim_{x \to a^{-}} \frac{f (x) - f (a)}{x - a} [L e f t d e r i v a t i v e]$
$lim_{x \to a^{+}} \frac{f (x) - f (a)}{x - a} [R i g h t d e r i v a t i v e]$

Step 3
If the one-sided derivatives are equal, then the function has a derivative at $x = a, t \hat{} i s f^{'} (x) = lim_{x \to^{-}} \frac{f (x) - f (a)}{x - a} = lim_{x \to^{+}} \frac{f (x) - f (a)}{x - a}$

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