Suppose I have a column matrix $x = \begin{pmatrix}x1\\x2\\x3 \end{pmatrix}$ and another column matrix $y = \begin{pmatrix}y1\\y2\\y3 \end{pmatrix}$. I know all the partial derivatives of $y$ with respect to $x$. Lets say they are $$\frac{\partial}{\partial x_i}y = \begin{pmatrix}y'_i1\\y'_i2\\y'_i3\end{pmatrix}$$ for $i \leq 3$.
Now my question is how can I get $\frac{d}{dx} (y)$, i.e., I want to differentiate $y$ with respect to $x$. How can I get that?
P.S. I am not a mathematics major so this may be a very lame question. But I want to know that. This is actually a subproblem that I need to solve for my original problem.
If $$\mathbf{y}=\begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_m \\ \end{bmatrix} $$ and $$ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \\ \end{bmatrix} $$ Then $$ \frac{\partial \mathbf{y}}{\partial \mathbf{x}} = \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} & \cdots & \frac{\partial y_1}{\partial x_n}\\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2} & \cdots & \frac{\partial y_2}{\partial x_n}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\partial y_m}{\partial x_1} & \frac{\partial y_m}{\partial x_2} & \cdots & \frac{\partial y_m}{\partial x_n}\\ \end{bmatrix}. $$