Sign of the U-Derivative

The sign of the directional derivative $D_\mathbf{u} f(\mathbf{a}) = \nabla f(\mathbf{a}) \cdot \mathbf{u}$ determines whether $f$ increases or decreases when we move from $\mathbf{a}$ in direction $\mathbf{u}$:

• $D_\mathbf{u} f > 0$: $f$ is increasing in direction $\mathbf{u}$ — moving in direction $\mathbf{u}$ increases $f$.
• $D_\mathbf{u} f < 0$: $f$ is decreasing in direction $\mathbf{u}$ — moving in direction $\mathbf{u}$ decreases $f$.
• $D_\mathbf{u} f = 0$: $f$ is neither increasing nor decreasing in direction $\mathbf{u}$ — moving tangentially along a level curve.

By the Cauchy-Schwarz inequality, $|\nabla f(\mathbf{a}) \cdot \mathbf{u}| \leq \|\nabla f(\mathbf{a})\|\|\mathbf{u}\| = \|\nabla f(\mathbf{a})\|$. Equality holds when $\mathbf{u} = \pm \nabla f(\mathbf{a})/\|\nabla f(\mathbf{a})\|$ — so the gradient direction gives the maximum directional derivative.