This post clears up a confusion I ran into recently with directional semi-derivatives in one dimension.
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
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.
- Wikipedia. Directional derivative.
- Wikipedia. Semi-differentiability.