Overview
In the numerator layout, if the numerator is a column vector and the deonominator is a scalar, the result is a column vector. That is, the derivative mimics the vector status of the numerator.
{%
\vec{x} =
\begin{bmatrix}
x_1 \\
x_2 \\
x_3 \\
\end{bmatrix}
%}
Then we have
{% \frac{d \vec{x}}{dt} =
\begin{bmatrix}
\frac{x_1}{dt} \\
\frac{x_2}{dt} \\
\frac{x_3}{dt} \\
\end{bmatrix}
%}
If we have another vector {% \vec{y} %} then we have
{%
\vec{y} =
\begin{bmatrix}
y_1 \\
y_2 \\
y_3 \\
\end{bmatrix}
%}
Then
{% \frac{dt}{d\vec{y}} =
\begin{bmatrix}
\frac{dt}{dy_1} &
\frac{dt}{dy_2} &
\frac{dt}{y_3} \\
\end{bmatrix}
%}
That is, if the column vector is in the denominator, the derivative is then a row vector.
Lastly, this allows us to write the derivative of one column vector by another column vector as a matrix.
{% \frac{d\vec{x}}{d\vec{y}} =
\begin{bmatrix}
\frac{dx_1}{dy_1} &\frac{dx_1}{dy_2} & \frac{dx_1}{dy_3} \\
\frac{dx_2}{dy_1} &\frac{dx_2}{dy_2} & \frac{dx_2}{dy_3} \\
\frac{dx_3}{dy_1} &\frac{dx_3}{dy_2} & \frac{dx_3}{dy_3} \\
\end{bmatrix}
%}
Formulas
{% \frac{\partial \vec{a}^T \vec{x}}{\partial \vec{x}} = \vec{a}^T %}
{% \frac{\partial \textbf{M} \vec{x}}{\partial \vec{x}} = \textbf{M} %}
Chain Rule
Given a vector valued set of functions
{% \vec{v} = f(g(h(\vec{x}))) %}
The derivative can be calculated as
{% \frac{d \vec{v}}{d \vec{x}} = \frac{df}{dg} \frac{dg}{dh} \frac{dh}{d\vec{x}} %}