U-Derivative for Vector-Valued Functions

For a vector-valued function $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^m$, the directional derivative in direction $\mathbf{u}$ at $\mathbf{a}$ is the vector: $D_\mathbf{u}\mathbf{f}(\mathbf{a}) = \lim_{t\to 0}\frac{\mathbf{f}(\mathbf{a}+t\mathbf{u})-\mathbf{f}(\mathbf{a})}{t} = J\mathbf{f}(\mathbf{a})\mathbf{u} \in \mathbb{R}^m$

This is the Jacobian-vector product: apply the Jacobian matrix $J\mathbf{f}(\mathbf{a})$ to the direction vector $\mathbf{u}$. The result tells us how all $m$ output components change simultaneously when we perturb the input in direction $\mathbf{u}$.

Component-wise: $(D_\mathbf{u}\mathbf{f})_i = D_\mathbf{u} f_i = \nabla f_i \cdot \mathbf{u}$ — the directional derivative of the $i$-th component. So the vector directional derivative is just a stack of scalar directional derivatives.