Clearing up directional semi-derivatives for one dimension

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

Directional semi-derivatives

Consider a function f: \mathbb{R}^d \mapsto \mathbb{R}, where d \geq 1, and let \mathbf{u} \in \mathbb{R}^d be a direction in \mathbb{R}^d (i.e. a vector with length 1). For any point \mathbf{x} \in \mathbb{R}^d, the directional semi-derivative of f at \mathbf{x} in the direction \mathbf{u} is given by

\begin{aligned} \nabla_{\mathbf{u}} f(\mathbf{x}) = \lim_{h \rightarrow 0^+} \dfrac{f(\mathbf{x} + h \mathbf{u}) - f(\mathbf{x})}{h}  \end{aligned}

if the limit exists. By h \rightarrow 0^+, we mean that h approaches zero from the positive direction. Intuitively, the directional semi-derivative says that if I were to move a small positive distance h from \mathbf{x} in the direction \mathbf{u}, f would increase by about h \nabla_{\mathbf{u}}f(\mathbf{x}).

In one dimension (d = 1), there are only two directions: \mathbf{u} = +1 and \mathbf{u} = -1. The formula above simplifies to

\begin{aligned} \nabla_{+1} f(x) = \lim_{h \rightarrow 0^+} \dfrac{f(x + h) - f(x)}{h}, \\  \nabla_{-1} f(x) = \lim_{h \rightarrow 0^+} \dfrac{f(x - h) - f(x)}{h}. \end{aligned}

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 f: \mathbb{R} \mapsto \mathbb{R}, the right and left derivatives of f at x are defined respectively as

\begin{aligned} \partial_+ f(x) &= \lim_{x' \rightarrow x^+} \dfrac{f(x') - f(x)}{x' - x}, \\  \partial_- f(x) &= \lim_{x' \rightarrow x^-} \dfrac{f(x') - f(x)}{x' - x}. \end{aligned}

Writing x' = x+h in the first equation and x' = x - h in the second equation, we can express the one-sided derivatives equivalently as

\begin{aligned} \partial_+ f(x) &= \lim_{h \rightarrow 0^+} \dfrac{f(x+h) - f(x)}{h}, \\  \partial_- f(x) &= \lim_{h \rightarrow 0^+} \dfrac{f(x - h) - f(x)}{-h}. \end{aligned}

Notice that the directional semi-derivative matches the one-sided derivative for the positive direction but not for the negative direction! We have \nabla_{+1} f(x) = \partial_+ f(x) but \nabla_{-1} f(x) = -\partial_- f(x). Thus, one needs to be clear when defining these quantities in one dimension to avoid confusion.


  1. Wikipedia. Directional derivative.
  2. Wikipedia. Semi-differentiability.

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