This is from one of the questions on stackexchange. However, I would like to extend this to the following:
Can someone disprove/prove it and give me references to why it is true/untrue?
Any resources that would help me resolve/understand this would be much appreciated.
At real calculus, we say that a function $f:A\subset\mathbb{R}\rightarrow\mathbb{R}$ is differentiable at $x_{0}\in A$ if the limit $$\lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}$$If the limit exists, we define the derivative of f at a, $f'(a)$ by its value.
Generally, we say that a function $f:U\subset\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is differentiable at $a\in U$ if there is a matrix B, $m\times n$ satisfying $$\frac{f(a+h)-f(a)-B\cdot h}{|h|} \rightarrow 0$$ when $|h|\rightarrow 0$. So, $B$ is the derivative of $f$ at $a$. Writting the usual notation, $B=Df(a)$. You can prove the dot product generally.
By the way, writting $\vec{x}\in\mathbb{R}^{n}$ by $\vec{x}=(x_{1},\dots,x_{n})$ and $f=(f_{1},\dots,f_{m}),$ $$Df(\vec{x})=\left[\begin{array}{ll} \frac{\partial f_{1}}{\partial x_{1}} \dots \frac{\partial f_{1}}{\partial x_{n}} \\ \,\,\,\,\vdots\,\,\,\,\,\,\,\,\,\,\,\,\,\vdots \\ \frac{\partial f_{m}}{\partial x_{1}} \dots \frac{\partial f_{m}}{\partial x_{n}} \end{array}\right] $$.
Let $k\leq n$ an integer and $\vec{z}\in\mathbb{R}^{k}$. We can write $\vec{x}=(\vec{z},\vec{w})$, where $\vec{w}\in\mathbb{R}^{n-k}$. So, $$Df(\vec{x})=\left[\begin{array}{ll} \frac{\partial f_{1}}{\partial \vec{z}}\,\,\frac{\partial f_{1}}{\partial \vec{w}} \\ \,\,\,\,\vdots\,\,\,\,\,\,\,\,\vdots \\ \frac{\partial f_{m}}{\partial \vec{z}}\,\,\frac{\partial f_{m}}{\partial \vec{w}} \end{array}\right] $$.
Thinking that way you are able to solve your problem