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Math101Directional Derivatives
A rigorous, example-driven guide to directional derivatives, including hypotheses, method choice, verification, and practice.
The central idea
If $f:\mathbb R^n\to\mathbb R$ is differentiable at $\mathbf a$, its directional derivative in a unit direction $\mathbf u$ is $D_{\mathbf u}f(\mathbf a)=\lim_{h\to0}[f(\mathbf a+h\mathbf u)-f(\mathbf a)]/h =\nabla f(\mathbf a)\cdot\mathbf u$. A nonunit direction changes the rate's scale.
Definitions, hypotheses, and notation
The formula from the gradient requires differentiability. Existence of all directional derivatives alone does not guarantee differentiability or a single linear map controlling every direction. At differentiable points, dependence on $\mathbf u$ is linear before unit normalization and the Cauchy–Schwarz inequality proves the maximum-rate statement.
A phrase such as 'toward point B' means subtract base point A from B; reversing the subtraction changes the derivative's sign. If coordinates have physical units or unequal scaling, the meaning of a Euclidean unit direction should be interpreted in that coordinate model.
Conceptual meaning
The gradient projects onto the chosen direction. Its largest possible directional derivative is $\|\nabla f\|$, attained along the gradient; the most negative is its opposite. Directions tangent to a level surface give zero first-order change.
A dependable method and decision rule
- Compute the displacement direction from the given geometric information.
- Normalize it to length one unless the problem explicitly defines otherwise.
- Compute and evaluate the gradient at the base point.
- Take the dot product.
- Interpret sign and units as change per unit distance in that direction.
