Total Derivative

$$Df=\left[\begin{array}{ccc}\frac{\partial }{\partial x_1}{f}_1& \cdots & \frac{\partial }{\partial x_n}{f}_1\\ ⋮& \ddots & ⋮\\ \frac{\partial }{\partial x_1}{f}_m& \cdots & \frac{\partial }{\partial x_n}{f}_m\end{array}\right]$$ $$f:{\mathbb{R}}^n\to {\mathbb{R}}^{m}f(x_1,\cdots ,x_n)=\left[\begin{array}{c}{f}_1(x_1,\cdots ,x_n)\\ ⋮\\ f_m(x_1,\cdots ,x_n))\end{array}\right]$$ $$f:{\mathbb{R}}^n\to {\mathbb{R}}^m⟹Df\in {\mathbb{R}}^{m\times n}$$

Examples

$$f(x)⟹Df=[f_x]$$ $$\vec{r}(x(t),y(t),z(t))\in {\mathbb{R}}^3⟹D_{\vec{r}}=\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]$$ $$f(x,y)⟹Df=\left[\begin{array}{cc}{f}_x& f_y\end{array}\right]$$ $$f(x,y,z)⟹Df=\left[\begin{array}{ccc}{f}_x& f_y& f_z\end{array}\right]$$ $$\vec{f}(x(s,t),y(s,t),z(s,t))⟹Df=\left[\begin{array}{cc}{x}_s& x_t\\ y_s& y_t\\ z_s& z_t\end{array}\right]$$