Directional Derivatives and the Gradient

Core Idea

The directional derivative D_u f = ∇f · u gives the rate of change in direction u (unit vector). The gradient ∇f = ⟨f_x, f_y⟩ points in the direction of steepest ascent and has magnitude equal to the maximum directional derivative.

Explainer

Partial derivatives tell you how fast f(x, y) changes when you move parallel to the x-axis or y-axis. But what if you walk diagonally, or in some arbitrary direction? The directional derivative answers the general question: how fast is f changing as I move in direction u? The answer turns out to be encoded entirely in the gradient ∇f, which you've already computed, combined with the dot product, which extracts components.

For a unit vector u = ⟨a, b⟩, the directional derivative is D_u f = ∇f · u = f_x · a + f_y · b. This formula says: project the gradient onto your direction of travel and read off the rate of change. Geometrically, the gradient ∇f = ⟨f_x, f_y⟩ is the "slope vector" of the surface — it captures all rate-of-change information in every direction simultaneously, and dotting with u extracts the slice relevant to your particular direction.

The deepest consequence follows from the geometry of dot products: D_u f = ‖∇f‖ cos θ, where θ is the angle between ∇f and u. This is maximized when θ = 0 — when you walk in the direction of ∇f itself. So the gradient points in the direction of steepest ascent, and ‖∇f‖ equals the maximum rate of increase. Walking in the direction of −∇f gives steepest descent. Walking perpendicular to ∇f gives D_u f = 0 — no change — meaning you're moving along a level curve where f is constant. The gradient is always perpendicular to the level curves of f.

To ground this concretely: take f(x, y) = x² + y² (a bowl-shaped paraboloid). At the point (1, 1), ∇f = ⟨2, 2⟩. Walking due east (u = ⟨1, 0⟩) gives D_u f = 2. Walking northeast in the gradient direction (u = ⟨1, 1⟩/√2) gives D_u f = ‖⟨2, 2⟩‖ = 2√2, a steeper climb. This is exactly why gradient descent algorithms — used throughout optimization and machine learning — follow −∇f to find function minima: the gradient tells you the direction of fastest increase, so its negative points most efficiently downhill.