derivatives of a vector of functions with respect to a vector Let $\vec W \in \mathbb R^3$. What is the general solution to:
$$\frac{\partial}{\partial \vec{W}} \begin{pmatrix}
        f(\vec W) \\
        g(\vec W)
        \end{pmatrix}
$$
I think that in the case where $f$ and $g$ are linear I could rewrite:
$$\begin{pmatrix}
        f(\vec W) \\
        g(\vec W)
        \end{pmatrix}
=A\cdot \vec W
$$
for some suitable matrix $A$ and then this would break down to:
$$\frac{\partial}{\partial \vec{W}}A\cdot \vec W=A
$$
Is this correct? So my question really aims for the general case, i.e. when $f$ and $g$ are not necessarily linear.
It feels kind of wrong to take the derivatives along the rows given that $\vec W$ is a column-vector.
 A: I think I've found what I needed on wikipedia and it is actually quite simple:
$$
\frac{\partial \mathbf{y}}{\partial \mathbf{x}} =
\begin{bmatrix}
\frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n}\\
\frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n}\\
\vdots & \vdots & \ddots & \vdots\\
\frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n}\\
\end{bmatrix}
$$
Hence in my case this it would look like this:
$$
\frac{\partial}{\partial \vec{W}} \begin{pmatrix}
        f(\vec W) \\
        g(\vec W)
        \end{pmatrix} =
\begin{bmatrix}
\frac{\partial f(\vec W)}{\partial W_1} & \frac{\partial f(\vec W)}{\partial W_2} &  \frac{\partial f(\vec W)}{\partial W_3}\\
\frac{\partial g(\vec W)}{\partial W_1} & \frac{\partial g(\vec W)}{\partial W_2} &  \frac{\partial g(\vec W)}{\partial W_3}\\
\end{bmatrix}
$$
The fact that the vector is a vector of functions doesn't add anything special to it...
A: What you write is a (dangerous) abuse of notation. In dimensions superior to 1, the concept of derivative depends on the direction see wikipedia
In the particular case where it is linear, you can see that:
$\frac{d}{dt} f(\vec a + t \vec W) = \frac{d}{dt} (A \vec a + t A \vec w ) = A \vec w$
You can also look at the gradient (taking partial derivatives), which is probably more what you're after.
