Jacobian

Its a matrix of first-order partial derivatives. Used in back propagation.

$$\mathbf{J}=\left[\begin{array}{cccc}\frac{\partial f_1}{\partial x_1}& \frac{\partial f_1}{\partial x_2}& \cdots & \frac{\partial f_1}{\partial x_n}\\ \frac{\partial f_2}{\partial x_1}& \frac{\partial f_2}{\partial x_2}& \cdots & \frac{\partial f_2}{\partial x_n}\\ ⋮& ⋮& \ddots & ⋮\\ \frac{\partial f_m}{\partial x_1}& \frac{\partial f_m}{\partial x_2}& \cdots & \frac{\partial f_m}{\partial x_n}\end{array}\right]$$

From First Principles

$$\mathbf{J}\triangleq \underset{n\to 0}{\lim }\frac{f(\mathbf{x}+\mathbf{h})-f(\mathbf{x})}{\mathbf{h}}\in {\mathbb{R}}^{n\times m}$$