Chain Rule for Multivariable Vector Functions
📂Vector Analysis Chain Rule for Multivariable Vector Functions Theorem Let’s assume that two functions g : D ⊂ R m → R k \mathbf{g} : D \subset \mathbb{R}^{m} \to \mathbb{R}^{k} g : D ⊂ R m → R k f : g ( R k ) ⊂ R k → R n \mathbf{f} : \mathbf{g}(\mathbb{R}^{k}) \subset \mathbb{R}^{k} \to \mathbb{R}^{n} f : g ( R k ) ⊂ R k → R n differentiable . Then, the composition of these two functions F = f ∘ g : R m → R n \mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{n} F = f ∘ g : R m → R n (total) derivative of F \mathbf{F} F
F ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x )
\mathbf{F}^{\prime}(\mathbf{x}) = \mathbf{f}^{\prime}\left( \mathbf{g}(\mathbf{x}) \right) \mathbf{g}^{\prime}(\mathbf{x})
F ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x )
Explanation This is called the chain rule .
If we denote x = ( x 1 , … , x m ) \mathbf{x} = (x_{1}, \dots, x_{m}) x = ( x 1 , … , x m ) g ( x ) = ( g 1 , … , g k ) \mathbf{g}(\mathbf{x}) = (g_{1}, \dots, g_{k}) g ( x ) = ( g 1 , … , g k ) f ( g 1 , … , g k ) = ( f 1 , … , f n ) \mathbf{f}(g_{1}, \dots, g_{k}) = (f_{1}, \dots, f_{n}) f ( g 1 , … , g k ) = ( f 1 , … , f n ) n × m n \times m n × m total derivative .
F ′ ( x ) = [ ∂ f 1 ( g ( x ) ) ∂ g 1 ∂ f 1 ∂ g 2 ⋯ ∂ f 1 ∂ g k ∂ f 2 ∂ g 1 ∂ f 2 ∂ g 2 ⋯ ∂ f 2 ∂ g k ⋮ ⋮ ⋱ ⋮ ∂ f n ∂ g 1 ∂ f n ∂ g 2 ⋯ ∂ f n ∂ g k ] [ ∂ g 1 ( x ) ∂ x 1 ∂ g 1 ∂ x 2 ⋯ ∂ g 1 ∂ x m ∂ g 2 ∂ x 1 ∂ g 2 ∂ x 2 ⋯ ∂ g 2 ∂ x m ⋮ ⋮ ⋱ ⋮ ∂ g k ∂ x 1 ∂ g k ∂ x 2 ⋯ ∂ g k ∂ x m ] = [ ∂ f 1 ∂ g 1 ∂ g 1 ∂ x 1 + ∂ f 1 ∂ g 2 ∂ g 2 ∂ x 1 + ⋯ + ∂ f 1 ∂ g k ∂ g k ∂ x 1 … ∂ f 1 ∂ g 1 ∂ g 1 ∂ x 1 + ∂ f 1 ∂ g 2 ∂ g 2 ∂ x m + ⋯ + ∂ f 1 ∂ g k ∂ g k ∂ x m ⋮ ⋱ ⋮ ∂ f n ∂ g 1 ∂ g 1 ∂ x 1 + ∂ f n ∂ g 2 ∂ g 2 ∂ x 1 + ⋯ + ∂ f n ∂ g k ∂ g k ∂ x 1 ⋯ ∂ f n ∂ g 1 ∂ g 1 ∂ x 1 + ∂ f n ∂ g 2 ∂ g 2 ∂ x m + ⋯ + ∂ f n ∂ g m ∂ g k ∂ x m ] = [ ∑ ℓ = 1 k ∂ f 1 ∂ g ℓ ∂ g ℓ ∂ x 1 … ∑ ℓ = 1 k ∂ f 1 ∂ g ℓ ∂ g ℓ ∂ x m ⋮ ⋱ ⋮ ∑ ℓ = 1 k ∂ f n ∂ g ℓ ∂ g ℓ ∂ x 1 … ∑ ℓ = 1 k ∂ f n ∂ g ℓ ∂ g ℓ ∂ x m ]
\begin{align*}
\mathbf{F}^{\prime} (\mathbf{x}) =&\ \begin{bmatrix}
\dfrac{\partial f_{1}(\mathbf{g}(\mathbf{x}))}{\partial g_{1}} & \dfrac{\partial f_{1}}{\partial g_{2}} & \cdots & \dfrac{\partial f_{1}}{\partial g_{k}}
\\[1em] \dfrac{\partial f_{2}}{\partial g_{1}} & \dfrac{\partial f_{2}}{\partial g_{2}} & \cdots & \dfrac{\partial f_{2}}{\partial g_{k}}
\\[1em] \vdots & \vdots & \ddots & \vdots
\\[1em] \dfrac{\partial f_{n}}{\partial g_{1}} & \dfrac{\partial f_{n}}{\partial g_{2}} & \cdots & \dfrac{\partial f_{n}}{\partial g_{k}}
\end{bmatrix}
\begin{bmatrix}
\dfrac{\partial g_{1}(\mathbf{x})}{\partial x_{1}} & \dfrac{\partial g_{1}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{1}}{\partial x_{m}}
\\[1em] \dfrac{\partial g_{2}}{\partial x_{1}} & \dfrac{\partial g_{2}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{2}}{\partial x_{m}}
\\[1em] \vdots & \vdots & \ddots & \vdots
\\[1em] \dfrac{\partial g_{k}}{\partial x_{1}} & \dfrac{\partial g_{k}}{\partial x_{2}} & \cdots & \dfrac{\partial g_{k}}{\partial x_{m}}
\end{bmatrix}
\\[1em] =&\ \begin{bmatrix}
\dfrac{\partial f_{1}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{1}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{1}} + \cdots + \dfrac{\partial f_{1}}{\partial g_{k}} \dfrac{\partial g_{k}}{\partial x_{1}} & \dots & \dfrac{\partial f_{1}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{1}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{m}} + \cdots + \dfrac{\partial f_{1}}{\partial g_{k}} \dfrac{\partial g_{k}}{\partial x_{m}}
\\[1em] \vdots & \ddots & \vdots
\\[1em] \dfrac{\partial f_{n}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{n}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{1}} + \cdots + \dfrac{\partial f_{n}}{\partial g_{k}} \dfrac{\partial g_{k}}{\partial x_{1}} & \cdots & \dfrac{\partial f_{n}}{\partial g_{1}} \dfrac{\partial g_{1}}{\partial x_{1}}+\dfrac{\partial f_{n}}{\partial g_{2}}\dfrac{\partial g_{2}}{\partial x_{m}} + \cdots + \dfrac{\partial f_{n}}{\partial g_{m}} \dfrac{\partial g_{k}}{\partial x_{m}}
\end{bmatrix}
\\[1em] =&\ \begin{bmatrix}
\displaystyle \sum\limits_{\ell =1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \displaystyle \sum\limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\\[1em] \vdots & \ddots & \vdots
\\[1em] \displaystyle \sum\limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \displaystyle \sum\limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\end{bmatrix}
\end{align*}
F ′ ( x ) = = = ∂ g 1 ∂ f 1 ( g ( x )) ∂ g 1 ∂ f 2 ⋮ ∂ g 1 ∂ f n ∂ g 2 ∂ f 1 ∂ g 2 ∂ f 2 ⋮ ∂ g 2 ∂ f n ⋯ ⋯ ⋱ ⋯ ∂ g k ∂ f 1 ∂ g k ∂ f 2 ⋮ ∂ g k ∂ f n ∂ x 1 ∂ g 1 ( x ) ∂ x 1 ∂ g 2 ⋮ ∂ x 1 ∂ g k ∂ x 2 ∂ g 1 ∂ x 2 ∂ g 2 ⋮ ∂ x 2 ∂ g k ⋯ ⋯ ⋱ ⋯ ∂ x m ∂ g 1 ∂ x m ∂ g 2 ⋮ ∂ x m ∂ g k ∂ g 1 ∂ f 1 ∂ x 1 ∂ g 1 + ∂ g 2 ∂ f 1 ∂ x 1 ∂ g 2 + ⋯ + ∂ g k ∂ f 1 ∂ x 1 ∂ g k ⋮ ∂ g 1 ∂ f n ∂ x 1 ∂ g 1 + ∂ g 2 ∂ f n ∂ x 1 ∂ g 2 + ⋯ + ∂ g k ∂ f n ∂ x 1 ∂ g k … ⋱ ⋯ ∂ g 1 ∂ f 1 ∂ x 1 ∂ g 1 + ∂ g 2 ∂ f 1 ∂ x m ∂ g 2 + ⋯ + ∂ g k ∂ f 1 ∂ x m ∂ g k ⋮ ∂ g 1 ∂ f n ∂ x 1 ∂ g 1 + ∂ g 2 ∂ f n ∂ x m ∂ g 2 + ⋯ + ∂ g m ∂ f n ∂ x m ∂ g k ℓ = 1 ∑ k ∂ g ℓ ∂ f 1 ∂ x 1 ∂ g ℓ ⋮ ℓ = 1 ∑ k ∂ g ℓ ∂ f n ∂ x 1 ∂ g ℓ … ⋱ … ℓ = 1 ∑ k ∂ g ℓ ∂ f 1 ∂ x m ∂ g ℓ ⋮ ℓ = 1 ∑ k ∂ g ℓ ∂ f n ∂ x m ∂ g ℓ
In Einstein notation , for 1 ≤ i ≤ n 1 \le i \le n 1 ≤ i ≤ n 1 ≤ j ≤ m 1 \le j \le m 1 ≤ j ≤ m
F ′ = [ F i j ′ ] = [ ∂ f 1 ∂ g ℓ ∂ g ℓ ∂ x 1 … ∂ f 1 ∂ g ℓ ∂ g ℓ ∂ x m ⋮ ⋱ ⋮ ∂ f n ∂ g ℓ ∂ g ℓ ∂ x 1 … ∂ f n ∂ g ℓ ∂ g ℓ ∂ x m ]
\mathbf{F}^{\prime} = \left[ F_{ij}^{\prime} \right] =
\begin{bmatrix}
\dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\\[1em] \vdots & \ddots & \vdots
\\[1em] \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\end{bmatrix}
F ′ = [ F ij ′ ] = ∂ g ℓ ∂ f 1 ∂ x 1 ∂ g ℓ ⋮ ∂ g ℓ ∂ f n ∂ x 1 ∂ g ℓ … ⋱ … ∂ g ℓ ∂ f 1 ∂ x m ∂ g ℓ ⋮ ∂ g ℓ ∂ f n ∂ x m ∂ g ℓ
F i j ′ = ∂ f i ∂ g ℓ ∂ g ℓ ∂ x j
F_{ij}^{\prime} = \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}}
F ij ′ = ∂ g ℓ ∂ f i ∂ x j ∂ g ℓ
Since this is the most generalized form, various specific formulas can be obtained according to k , m , n k, m, n k , m , n
Case 1. g : R → R g : \mathbb{R} \to \mathbb{R} g : R → R f : R → R f : \mathbb{R} \to \mathbb{R} f : R → R F = f ∘ g : R → R F = f \circ g : \mathbb{R} \to \mathbb{R} F = f ∘ g : R → R
When x ∈ R x \in \mathbb{R} x ∈ R g = g ( x ) g = g(x) g = g ( x ) f = f ( g ( x ) ) f = f(g(x)) f = f ( g ( x ))
F ′ = d F d x = d f d g d g d x
F^{\prime} = \dfrac{d F}{d x} = \dfrac{d f}{d g} \dfrac{d g}{d x}
F ′ = d x d F = d g df d x d g
Proof
Case 2. g : R → R k \mathbf{g} : \mathbb{R} \to \mathbb{R}^{k} g : R → R k f : R k → R f : \mathbb{R}^{k} \to \mathbb{R} f : R k → R F = f ∘ g : R → R F = f \circ \mathbf{g} : \mathbb{R} \to \mathbb{R} F = f ∘ g : R → R
When x ∈ R x \in \mathbb{R} x ∈ R g ( x ) = ( g 1 , … , g k ) \mathbf{g}(x) = (g_{1}, \dots, g_{k}) g ( x ) = ( g 1 , … , g k ) f = f ( g 1 , … , g k ) f = f(g_{1}, \dots ,g_{k}) f = f ( g 1 , … , g k )
F ′ = d F d x = ∑ ℓ = 1 k ∂ f ∂ g ℓ d g ℓ d x
F^{\prime} = \dfrac{d F}{d x} = \sum \limits_{\ell=1}^{k}\dfrac{\partial f}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x}
F ′ = d x d F = ℓ = 1 ∑ k ∂ g ℓ ∂ f d x d g ℓ
Case 3. g : R m → R g : \mathbb{R}^{m} \to \mathbb{R} g : R m → R f : R → R f : \mathbb{R} \to \mathbb{R} f : R → R F = f ∘ g : R m → R F = f \circ g : \mathbb{R}^{m} \to \mathbb{R} F = f ∘ g : R m → R
When x = ( x 1 , … , x n ) ∈ R n \mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} x = ( x 1 , … , x n ) ∈ R n g = g ( x ) g = g(\mathbf{x}) g = g ( x ) f = f ( g ( x ) ) f = f(g(\mathbf{x})) f = f ( g ( x ))
F ′ = d F d x = [ d f d g ∂ g ∂ x 1 … d f d g ∂ g ∂ x m ]
F^{\prime} = \dfrac{d F}{d \mathbf{x}} =
\begin{bmatrix}
\dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{m}}
\end{bmatrix}
F ′ = d x d F = [ d g df ∂ x 1 ∂ g … d g df ∂ x m ∂ g ]
F j ′ = d f d g ∂ g ∂ x j , 1 ≤ j ≤ m
F_{j}^{\prime} = \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{j}},\quad 1 \le j \le m
F j ′ = d g df ∂ x j ∂ g , 1 ≤ j ≤ m
Case 4. g : R m → R k \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{k} g : R m → R k f : R k → R f : \mathbb{R}^{k} \to \mathbb{R} f : R k → R F = f ∘ g : R m → R F = f \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R} F = f ∘ g : R m → R
When x = ( x 1 , … , x n ) ∈ R n \mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} x = ( x 1 , … , x n ) ∈ R n g ( x ) = ( g 1 , … , g k ) \mathbf{g}(x) = (g_{1}, \dots, g_{k}) g ( x ) = ( g 1 , … , g k ) f = f ( g 1 , … , g k ) f = f(g_{1}, \dots, g_{k}) f = f ( g 1 , … , g k )
F ′ = d F d x = [ ∑ ℓ = 1 k ∂ f ∂ g ℓ ∂ g ℓ ∂ x 1 … ∑ ℓ = 1 k ∂ f ∂ g ℓ ∂ g ℓ ∂ x m ]
F^{\prime} = \dfrac{d F}{d \mathbf{x}} =
\begin{bmatrix}
\sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\end{bmatrix}
F ′ = d x d F = [ ℓ = 1 ∑ k ∂ g ℓ ∂ f ∂ x 1 ∂ g ℓ … ℓ = 1 ∑ k ∂ g ℓ ∂ f ∂ x m ∂ g ℓ ]
F j ′ = ∑ ℓ = 1 k ∂ f ∂ g ℓ ∂ g ℓ ∂ x j , 1 ≤ j ≤ m
F_{j}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}},\quad 1 \le j \le m
F j ′ = ℓ = 1 ∑ k ∂ g ℓ ∂ f ∂ x j ∂ g ℓ , 1 ≤ j ≤ m
Case 5. g : R → R g : \mathbb{R} \to \mathbb{R} g : R → R f : R → R n \mathbf{f} : \mathbb{R} \to \mathbb{R}^{n} f : R → R n F = f ∘ g : R → R n \mathbf{F} = \mathbf{f} \circ g : \mathbb{R} \to \mathbb{R}^{n} F = f ∘ g : R → R n
When x ∈ R x \in \mathbb{R} x ∈ R g = g ( x ) g = g(x) g = g ( x ) f ( g ( x ) ) = ( f 1 , … , f n ) \mathbf{f}(g(x)) = (f_{1}, \dots, f_{n}) f ( g ( x )) = ( f 1 , … , f n )
F ′ = d F d x = [ d f 1 d g d g d x ⋮ d f n d g d g d x ]
\mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d x} =
\begin{bmatrix}
\dfrac{d f_{1}}{d g} \dfrac{d g}{d x}
\\[1em] \vdots
\\[1em] \dfrac{d f_{n}}{d g} \dfrac{d g}{d x}
\end{bmatrix}
F ′ = d x d F = d g d f 1 d x d g ⋮ d g d f n d x d g
F i ′ = d f i d g d g d x , 1 ≤ i ≤ n
F_{i}^{\prime} = \dfrac{d f_{i}}{d g} \dfrac{d g}{d x},\quad 1\le i \le n
F i ′ = d g d f i d x d g , 1 ≤ i ≤ n
Case 6. g : R → R k \mathbf{g} : \mathbb{R} \to \mathbb{R}^{k} g : R → R k f : R k → R n \mathbf{f} : \mathbb{R}^{k} \to \mathbb{R}^{n} f : R k → R n F = f ∘ g : R → R n \mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R} \to \mathbb{R}^{n} F = f ∘ g : R → R n
When x ∈ R x \in \mathbb{R} x ∈ R g ( x ) = ( g 1 , … , g k ) \mathbf{g}(x) = (g_{1}, \dots, g_{k}) g ( x ) = ( g 1 , … , g k ) f ( g 1 , … , g k ) = ( f 1 , … , f n ) \mathbf{f}(g_{1}, \dots ,g_{k}) = (f_{1}, \dots, f_{n}) f ( g 1 , … , g k ) = ( f 1 , … , f n )
F ′ = d F d x = [ ∑ ℓ = 1 k ∂ f 1 ∂ g ℓ d g ℓ d x ⋮ ∑ ℓ = 1 k ∂ f n ∂ g ℓ d g ℓ d x ]
\mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d x} =
\begin{bmatrix}
\sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x}
\\[1em] \vdots
\\[1em] \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x}
\end{bmatrix}
F ′ = d x d F = ℓ = 1 ∑ k ∂ g ℓ ∂ f 1 d x d g ℓ ⋮ ℓ = 1 ∑ k ∂ g ℓ ∂ f n d x d g ℓ
F i ′ = ∑ ℓ = 1 k ∂ f i ∂ g ℓ d g ℓ d x , 1 ≤ i ≤ n
F_{i}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x},\quad 1\le i \le n
F i ′ = ℓ = 1 ∑ k ∂ g ℓ ∂ f i d x d g ℓ , 1 ≤ i ≤ n
Case 7. g : R m → R g : \mathbb{R}^{m} \to \mathbb{R} g : R m → R f : R → R n \mathbf{f} : \mathbb{R} \to \mathbb{R}^{n} f : R → R n F = f ∘ g : R m → R n \mathbf{F} = \mathbf{f} \circ g : \mathbb{R}^{m} \to \mathbb{R}^{n} F = f ∘ g : R m → R n
When x = ( x 1 , … , x n ) ∈ R n \mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} x = ( x 1 , … , x n ) ∈ R n g = g ( x ) g = g(\mathbf{x}) g = g ( x ) f ( g ( x ) ) = ( f 1 , … , f n ) \mathbf{f}(g(\mathbf{x})) = (f_{1}, \dots, f_{n}) f ( g ( x )) = ( f 1 , … , f n )
F ′ = d F d x = [ d f 1 d g ∂ g ∂ x 1 … d f 1 d g ∂ g ∂ x m ⋮ ⋱ ⋮ d f n d g ∂ g ∂ x 1 … d f n d g ∂ g ∂ x m ]
\mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d \mathbf{x}} =
\begin{bmatrix}
\dfrac{d f_{1}}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f_{1}}{d g} \dfrac{\partial g}{\partial x_{m}}
\\[1em] \vdots & \ddots & \vdots
\\[1em] \dfrac{d f_{n}}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f_{n}}{d g} \dfrac{\partial g}{\partial x_{m}}
\end{bmatrix}
F ′ = d x d F = d g d f 1 ∂ x 1 ∂ g ⋮ d g d f n ∂ x 1 ∂ g … ⋱ … d g d f 1 ∂ x m ∂ g ⋮ d g d f n ∂ x m ∂ g
F i j ′ = d f i d g ∂ g ∂ x j , 1 ≤ i ≤ n , 1 ≤ j ≤ m
F_{ij}^{\prime} = \dfrac{d f_{i}}{d g} \dfrac{\partial g}{\partial x_{j}},\quad 1\le i \le n, 1 \le j \le m
F ij ′ = d g d f i ∂ x j ∂ g , 1 ≤ i ≤ n , 1 ≤ j ≤ m
Case 8. g : R m → R k \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{k} g : R m → R k f : R k → R n \mathbf{f} : \mathbb{R}^{k} \to \mathbb{R}^{n} f : R k → R n F = f ∘ g : R m → R n \mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{n} F = f ∘ g : R m → R n
When x = ( x 1 , … , x n ) ∈ R n \mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n} x = ( x 1 , … , x n ) ∈ R n g ( x ) = ( g 1 , … , g k ) g(\mathbf{x}) = (g_{1}, \dots, g_{k}) g ( x ) = ( g 1 , … , g k ) f ( g 1 , … , g k ) = ( f 1 , … , f n ) \mathbf{f}(g_{1}, \dots, g_{k}) = (f_{1}, \dots, f_{n}) f ( g 1 , … , g k ) = ( f 1 , … , f n )
F ′ = d F d x = [ ∑ ℓ = 1 k ∂ f 1 ∂ g ℓ ∂ g ℓ ∂ x 1 … ∑ ℓ = 1 k ∂ f 1 ∂ g ℓ ∂ g ℓ ∂ x m ⋮ ⋱ ⋮ ∑ ℓ = 1 k ∂ f n ∂ g ℓ ∂ g ℓ ∂ x 1 … ∑ ℓ = 1 k ∂ f n ∂ g ℓ ∂ g ℓ ∂ x m ]
\mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d \mathbf{x}} =
\begin{bmatrix}
\sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\\[1em] \vdots & \ddots & \vdots
\\[1em] \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}}
\end{bmatrix}
F ′ = d x d F = ℓ = 1 ∑ k ∂ g ℓ ∂ f 1 ∂ x 1 ∂ g ℓ ⋮ ℓ = 1 ∑ k ∂ g ℓ ∂ f n ∂ x 1 ∂ g ℓ … ⋱ … ℓ = 1 ∑ k ∂ g ℓ ∂ f 1 ∂ x m ∂ g ℓ ⋮ ℓ = 1 ∑ k ∂ g ℓ ∂ f n ∂ x m ∂ g ℓ
F i j ′ = ∑ ℓ = 1 k ∂ f i ∂ g ℓ ∂ g ℓ ∂ x j , 1 ≤ i ≤ n , 1 ≤ j ≤ m
F_{ij}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}},\quad 1\le i \le n, 1 \le j \le m
F ij ′ = ℓ = 1 ∑ k ∂ g ℓ ∂ f i ∂ x j ∂ g ℓ , 1 ≤ i ≤ n , 1 ≤ j ≤ m
Proof Refer to the generalized proof .
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