A **one sided derivative** is either a *derivative from the left* or a *derivative from the right*.

**Derivative from the left**: You approach a point from the left direction of the number line.**Derivative from the right:**You approach a point from the right direction of the number line.

These are particularly useful at endpoints, where a function stops abruptly and doesn’t go beyond a certain point.

Any function that is differentiable at the end of its domain is called

*one sided differentiable*(Reinholz, n.d.).

Note though, that if both the right and left hand derivatives are equal, the derivative is an **ordinary derivative**, not a one sided derivative. Ordinary derivatives are the ones you’re normally used to dealing with in calculus; Another way to define them is that they are *not* partial derivatives.

## A More Formal Definition of a One Sided Derivative

A one sided derivative can be defined more formally as (Fogel, n.d.):

If f is a function on a half closed interval **[a, b)**, then:

The **right hand derivative** at a, denoted f′_{–} is the number

If it exists,

And

If the function is also defined on a half closed interval

**(a, b],** then:

The **left hand derivative **at b, denoted f′_{+} is the number

If it exists.

## Connection with Limits

In a practical sense, one sided derivatives are analogous to one sided limits.

See also: One sided limits.

## References

Aramanovich, I. et al. (2014). http://Mathematical Analysis: Differentiation and Integration [Print Replica]. Pergamon.

Fogel, M. The Derivative. Retrieved December 29, 2019 from: http://staff.imsa.edu/~fogel/Analysis/PDF/25%20The%20Derivative

Hazelwinkle, M. (1990). Encyclopedia of Mathematics. Springer.

Math Boys. Absolute Value Function. Retrieved August 2019 from: https://www.statisticshowto.com/absolute-value-function/

Reinholz, D. Derivatives. Retrieved December 29, 2019 from: https://www.ocf.berkeley.edu/~reinholz/ed/08sp_m160/lectures/derivatives.pdf