Differentiation Rules for Real Vector Functions

Theorem: Linearity of Differentiation

Let $f,g:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be real vector functions.

If $f$ and $g$ are both differentiable at $\mathbf{a}\in \mathcal{D}$, then for all $\lambda ,\mu \in \mathbb{R}$, the [function] $\lambda f+\mu g$ is also differentiable there with

$$\mathrm{d}(\lambda f+\mu g{)}_{\mathbf{a}}=\lambda \mathrm{d}{f}_{\mathbf{a}}+\mu \mathrm{d}{g}_{\mathbf{a}}$$

PROOF

TODO

Theorem: Chain Rule

Let $g:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$ be a real vector function on an open set $\mathcal{D}$ such that the image $g(\mathcal{D})$ is also open and let $f:g(\mathcal{D})\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^p$.

If $g$ is differentiable at $\mathbf{a}\in \mathcal{D}$ and $f$ is differentiable at $g(\mathbf{a})$, then the composition $f\circ g:\mathcal{D}\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^p$ is also differentiable at $\mathbf{a}$ with

$$\mathrm{d}(f\circ g{)}_{\mathbf{a}}=\mathrm{d}{f}_{\mathbf{g}(\mathbf{a})}\circ \mathrm{d}{g}_{\mathbf{a}}$$

PROOF

TODO