U-Derivative of a Linear Function

For a linear function $f(\mathbf{x}) = \mathbf{c}^T \mathbf{x}$, the directional derivative in any direction $\mathbf{u}$ is simply $D_\mathbf{u} f(\mathbf{a}) = \mathbf{c}^T \mathbf{u}$, regardless of the base point $\mathbf{a}$.

This follows directly from the definition: $\frac{f(\mathbf{a}+t\mathbf{u}) - f(\mathbf{a})}{t} = \frac{\mathbf{c}^T(\mathbf{a}+t\mathbf{u}) - \mathbf{c}^T\mathbf{a}}{t} = \frac{t\mathbf{c}^T\mathbf{u}}{t} = \mathbf{c}^T\mathbf{u}$.

For a linear function, the directional derivative equals the dot product of the coefficient vector with the direction. This is both a check of the formula and a preview of the general result: for any differentiable $f$, $D_\mathbf{u} f(\mathbf{a}) = \nabla f(\mathbf{a}) \cdot \mathbf{u}$.