This post clears up a confusion I ran into recently with directional semi-derivatives in one dimension.

**Directional semi-derivatives**

Consider a function , where , and let be a direction in (i.e. a vector with length 1). For any point , the **directional semi-derivative** of at in the direction is given by

if the limit exists. By , we mean that approaches zero from the positive direction. Intuitively, the directional semi-derivative says that if I were to move a small positive distance from in the direction , would increase by about .

In one dimension (), there are only two directions: and . The formula above simplifies to

**One-sided derivatives**

Some introductory courses on single-variable calculus introduce the concept of a **one-sided derivative**, also called **left and right derivatives**. For a function , the right and left derivatives of at are defined respectively as

Writing in the first equation and in the second equation, we can express the one-sided derivatives equivalently as

**Notice that the directional semi-derivative matches the one-sided derivative for the positive direction but not for the negative direction!** We have but . Thus, one needs to be clear when defining these quantities in one dimension to avoid confusion.

References:

- Wikipedia. Directional derivative.
- Wikipedia. Semi-differentiability.

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