Foundations — Directional derivative — definition, formula

This page builds the alphabet the parent Directional derivative note secretly assumes. Read top to bottom: each symbol is earned before the next uses it.

1. A point $\mathbf{a}$ — where you stand

Before we can ask "how fast does height change," we need to say where we are standing. In flat land, a location needs two numbers: how far east ($x$) and how far north ($y$).

The picture is a single dot on a grid. Nothing moves yet — this is just "you are here."

2. A function $f$ — the height of the ground

Now attach a height to every location. That is what a function of two variables does.

The picture is a landscape: a bumpy surface floating above the flat grid. Each grid point pushes up (or down) to the surface by an amount $f(x,y)$.

Related vault reading: Level curves and level sets slices this surface at constant heights.

3. A direction $\mathbf{u}$ and its length $\Vert \mathbf{u}\Vert$ — the way you face

A vector is an arrow: it has a direction and a length. We write $\mathbf{u}=(u_1,u_2)$ — the same pair-of-numbers notation as a point, but now it means "go $u_1$ east and $u_2$ north from wherever you are."

4. Partial derivatives $\frac{\partial f}{\partial x}$ — steepness facing due east

Before we allow any direction, master the two easy ones: due east and due north.

The picture: slice the landscape with a vertical wall running east–west. The cut edge is an ordinary 1-D curve, and $f_x$ is its slope.

5. The gradient $\nabla f$ — the arrow pointing most uphill

Bundle the two compass slopes into one arrow.

6. The dot product $\nabla f\cdot \mathbf{u}$ — how much your facing leans toward uphill

Two arrows, one number.

7. The directional derivative $D_{\mathbf{u}}f$ — the one number all the symbols build toward

Now we can finally name the thing itself.

To make that number precise we need one more tool — the limit.

Here is the helper $g$ tied to our pieces. Standing at $\mathbf{a}$ and stepping a distance $h$ along the unit direction $\mathbf{u}$ lands you at the point $\mathbf{a}+h\mathbf{u}$. Let $g(h)=f(\mathbf{a}+h\mathbf{u})$ be the height as you walk the straight line in direction $\mathbf{u}$. Then $g(0)=f(\mathbf{a})$ (you have not moved), and the difference quotient is exactly "rise in height $÷$ distance walked." Feeding this $g$ into the limit gives the full definition every symbol on this page was building toward:

And this limit collapses (when $f$ is differentiable) into the clean shortcut using the gradient and dot product from §5–§6: $D_{\mathbf{u}}f=\nabla f\cdot \mathbf{u}=\Vert \nabla f\Vert \cos \theta ,$ which is largest ($\theta =0$, face straight uphill), zero ($\theta =90^{\circ }$, walk along a level curve), and most negative ($\theta =180^{\circ }$, straight downhill).

Prerequisite map

Equipment checklist

Test yourself — reveal only after you have answered aloud.