Derivative of a vector with Respect to scalar? I have a function $f=w\begin{bmatrix} a \\b\\c \end{bmatrix}$.
So what is the $\frac{\partial f}{\partial w}$? 
I have never seen this.
Thank you!
 A: Your $f$ is just a linear map! Its derivative is the constant function $$f' : \mathbb{R} \rightarrow \mathbb{R}^3, \; \; x \mapsto \begin{pmatrix} a \\ b \\ c \end{pmatrix}.$$
More generally if you have $f$ given as a function $f = \begin{pmatrix} f_1 \\ f_2 \\ f_3 \end{pmatrix}$ where $f_1, f_2, f_3 : \mathbb{R} \rightarrow \mathbb{R}$ are differentiable, then the derivative of $f$ will be $\begin{pmatrix} f_1' \\ f_2' \\ f_3' \end{pmatrix}.$
A: Note that $\vec{f} = \begin{bmatrix} aw \\bw\\cw \end{bmatrix}$.
Now, differentiate each component of the vector with respect to $w$.  That is:
$$\frac{\partial\vec{f}}{\partial w} = \begin{bmatrix} \frac{\partial}{\partial w}aw \\\frac{\partial}{\partial w}bw\\\frac{\partial}{\partial w}cw \end{bmatrix} = \begin{bmatrix} a \\b\\c \end{bmatrix}$$
This assumes $a, b, c$ could be constants or functions (but not of $w$).
A: Partial derivatives are directional (Gâteaux) derivatives, in other words, since $w \in \mathbb{R}$,
$$ \frac{\partial f_w}{\partial w} = \lim_{h \in \mathbb{R}, h\to 0} \frac{f_{w+h}-f_w}{h} $$
Therefore
$$
   \begin{aligned}
      \frac{\partial f_w}{\partial w} & = \lim_{h \to 0} \;\frac{1}{h}(w + h - w)\begin{pmatrix} a \\ b \\ c \end{pmatrix} \\
                                      & = \begin{pmatrix} a \\ b \\ c \end{pmatrix}
   \end{aligned}
$$
This is how it works, but using relevant theorems like @Cocopuffs did will obviously give you the answer much faster.
