Total derivative or Jacobian matrix
The total derivative of a function f : R m → R n \mathbf f : \mathbb R^m \to \mathbb R^n f : R m → R n n × m n \times m n × m n n n m m m
Formally for a function f ( x 1 , x 2 , . . . , x m ) = ( f 1 , f 2 , . . . , f n ) \mathbf f(x_1, x_2, ..., x_m) = (f_1, f_2, ..., f_n) f ( x 1 , x 2 , ... , x m ) = ( f 1 , f 2 , ... , f n )
D f = ( ∂ f 1 ∂ x 1 ∂ f 1 ∂ x 2 ⋯ ∂ f 1 ∂ x m ∂ f 2 ∂ x 1 ∂ f 2 ∂ x 2 ⋯ ∂ f 2 ∂ x m ⋮ ⋮ ⋱ ⋮ ∂ f n ∂ x 1 ∂ f n ∂ x 2 ⋯ ∂ f n ∂ x m )
Df = \begin{pmatrix}
\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} & \cdots & \frac{\partial f_1}{\partial x_m} \\
\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} & \cdots & \frac{\partial f_2}{\partial x_m} \\
\vdots & \vdots & \ddots & \vdots \\
\frac{\partial f_n}{\partial x_1} & \frac{\partial f_n}{\partial x_2} & \cdots & \frac{\partial f_n}{\partial x_m}
\end{pmatrix}
D f = ∂ x 1 ∂ f 1 ∂ x 1 ∂ f 2 ⋮ ∂ x 1 ∂ f n ∂ x 2 ∂ f 1 ∂ x 2 ∂ f 2 ⋮ ∂ x 2 ∂ f n ⋯ ⋯ ⋱ ⋯ ∂ x m ∂ f 1 ∂ x m ∂ f 2 ⋮ ∂ x m ∂ f n
Recall
∂ f i ∂ x j \frac{\partial f_i}{\partial x_j} ∂ x j ∂ f i f i f_i f i x j x_j x j
The total derivative is sometimes written as D f p D\mathbf f_{\mathbf p} D f p f ′ ( p ) \mathbf f'(\mathbf p) f ′ ( p ) p \mathbf p p