U-Derivative and the LLA

The local linear approximation gives us a powerful formula for directional derivatives. If $f$ is differentiable at $\mathbf{a}$, then near $\mathbf{a}$ we have $f(\mathbf{a}+t\mathbf{u}) \approx f(\mathbf{a}) + Df(\mathbf{a})(t\mathbf{u}) = f(\mathbf{a}) + t \cdot Df(\mathbf{a})\mathbf{u}$.

Substituting into the definition: $D_\mathbf{u} f(\mathbf{a}) = \lim_{t\to 0} \frac{f(\mathbf{a}+t\mathbf{u})-f(\mathbf{a})}{t} = \lim_{t\to 0} \frac{t \cdot Df(\mathbf{a})\mathbf{u} + o(t)}{t} = Df(\mathbf{a})\mathbf{u}$.

In terms of the gradient: $D_\mathbf{u} f(\mathbf{a}) = Df(\mathbf{a})\mathbf{u} = \nabla f(\mathbf{a}) \cdot \mathbf{u}$. This is the fundamental formula connecting the Jacobian, the gradient, and directional derivatives.