다변수 벡터함수의 연쇄법칙
정리
두 함수 $\mathbf{g} : D \subset \mathbb{R}^{m} \to \mathbb{R}^{k}$, $\mathbf{f} : \mathbf{g}(\mathbb{R}^{k}) \subset \mathbb{R}^{k} \to \mathbb{R}^{n}$가 미분 가능하다고 하자. 그러면 두 함수의 합성 $\mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{n}$도 미분가능하고, $\mathbf{F}$의 (전)도함수는 다음을 만족한다.
$$ \mathbf{F}^{\prime}(\mathbf{x}) = \mathbf{f}^{\prime}\left( \mathbf{g}(\mathbf{x}) \right) \mathbf{g}^{\prime}(\mathbf{x}) $$
설명
이를 연쇄법칙 이라 한다.
$\mathbf{x} = (x_{1}, \dots, x_{m})$, $\mathbf{g}(\mathbf{x}) = (g_{1}, \dots, g_{k})$, $\mathbf{f}(g_{1}, \dots, g_{k}) = (f_{1}, \dots, f_{n})$라고 하면, 공식의 구체적인 꼴은 전 도함수의 정의로부터 다음과 같은 $n \times m$ 행렬이다.
$$ \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*} $$
아인슈타인 표기법으로 간단히 나타내면, $1 \le i \le n$, $1 \le j \le m$에 대해서
$$ \mathbf{F}^{\prime} = \left[ F_{ij}^{\prime} \right] = \begin{bmatrix} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix} $$
$$ F_{ij}^{\prime} = \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}} $$
이는 가장 일반화된 꼴이므로, $k, m, n$에 따라서 여러가지 구체적은 공식을 얻을 수 있다.
공식
Case 1. $g : \mathbb{R} \to \mathbb{R}$, $f : \mathbb{R} \to \mathbb{R}$, $F = f \circ g : \mathbb{R} \to \mathbb{R}$
$x \in \mathbb{R}$, $g = g(x)$, $f = f(g(x))$일 때,
$$ F^{\prime} = \dfrac{d F}{d x} = \dfrac{d f}{d g} \dfrac{d g}{d x} $$
Case 2. $\mathbf{g} : \mathbb{R} \to \mathbb{R}^{k}$, $f : \mathbb{R}^{k} \to \mathbb{R}$, $F = f \circ \mathbf{g} : \mathbb{R} \to \mathbb{R}$
$x \in \mathbb{R}$, $\mathbf{g}(x) = (g_{1}, \dots, g_{k})$, $f = f(g_{1}, \dots ,g_{k})$일 때,
$$ F^{\prime} = \dfrac{d F}{d x} = \sum \limits_{\ell=1}^{k}\dfrac{\partial f}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x} $$
Case 3. $g : \mathbb{R}^{m} \to \mathbb{R}$, $f : \mathbb{R} \to \mathbb{R}$, $F = f \circ g : \mathbb{R}^{m} \to \mathbb{R}$
$\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}$, $g = g(\mathbf{x})$, $f = f(g(\mathbf{x}))$일 때,
$$ F^{\prime} = \dfrac{d F}{d \mathbf{x}} = \begin{bmatrix} \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{m}} \end{bmatrix} $$
$$ F_{j}^{\prime} = \dfrac{d f}{d g} \dfrac{\partial g}{\partial x_{j}},\quad 1 \le j \le m $$
Case 4. $\mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{k}$, $f : \mathbb{R}^{k} \to \mathbb{R}$, $F = f \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}$
$\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}$, $\mathbf{g}(x) = (g_{1}, \dots, g_{k})$, $f = f(g_{1}, \dots, g_{k})$일 때,
$$ F^{\prime} = \dfrac{d F}{d \mathbf{x}} = \begin{bmatrix} \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix} $$
$$ F_{j}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}},\quad 1 \le j \le m $$
Case 5. $g : \mathbb{R} \to \mathbb{R}$, $\mathbf{f} : \mathbb{R} \to \mathbb{R}^{n}$, $\mathbf{F} = \mathbf{f} \circ g : \mathbb{R} \to \mathbb{R}^{n}$
$x \in \mathbb{R}$, $g = g(x)$, $\mathbf{f}(g(x)) = (f_{1}, \dots, f_{n})$일 때,
$$ \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d x} = \begin{bmatrix} \dfrac{d f_{1}}{d g} \dfrac{d g}{d x} \\[1em] \vdots \\[1em] \dfrac{d f_{n}}{d g} \dfrac{d g}{d x} \end{bmatrix} $$
$$ F_{i}^{\prime} = \dfrac{d f_{i}}{d g} \dfrac{d g}{d x},\quad 1\le i \le n $$
Case 6. $\mathbf{g} : \mathbb{R} \to \mathbb{R}^{k}$, $\mathbf{f} : \mathbb{R}^{k} \to \mathbb{R}^{n}$, $\mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R} \to \mathbb{R}^{n}$
$x \in \mathbb{R}$, $\mathbf{g}(x) = (g_{1}, \dots, g_{k})$, $\mathbf{f}(g_{1}, \dots ,g_{k}) = (f_{1}, \dots, f_{n})$일 때,
$$ \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d x} = \begin{bmatrix} \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x} \\[1em] \vdots \\[1em] \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x} \end{bmatrix} $$
$$ F_{i}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{d g_{\ell}}{d x},\quad 1\le i \le n $$
Case 7. $g : \mathbb{R}^{m} \to \mathbb{R}$, $\mathbf{f} : \mathbb{R} \to \mathbb{R}^{n}$, $\mathbf{F} = \mathbf{f} \circ g : \mathbb{R}^{m} \to \mathbb{R}^{n}$
$\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}$, $g = g(\mathbf{x})$, $\mathbf{f}(g(\mathbf{x})) = (f_{1}, \dots, f_{n})$일 때,
$$ \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d \mathbf{x}} = \begin{bmatrix} \dfrac{d f_{1}}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f_{1}}{d g} \dfrac{\partial g}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \dfrac{d f_{n}}{d g} \dfrac{\partial g}{\partial x_{1}} & \dots & \dfrac{d f_{n}}{d g} \dfrac{\partial g}{\partial x_{m}} \end{bmatrix} $$
$$ F_{ij}^{\prime} = \dfrac{d f_{i}}{d g} \dfrac{\partial g}{\partial x_{j}},\quad 1\le i \le n, 1 \le j \le m $$
Case 8. $\mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{k}$, $\mathbf{f} : \mathbb{R}^{k} \to \mathbb{R}^{n}$, $\mathbf{F} = \mathbf{f} \circ \mathbf{g} : \mathbb{R}^{m} \to \mathbb{R}^{n}$
$\mathbf{x} = (x_{1}, \dots, x_{n}) \in \mathbb{R}^{n}$, $g(\mathbf{x}) = (g_{1}, \dots, g_{k})$, $\mathbf{f}(g_{1}, \dots, g_{k}) = (f_{1}, \dots, f_{n})$일 때,
$$ \mathbf{F}^{\prime} = \dfrac{d \mathbf{F}}{d \mathbf{x}} = \begin{bmatrix} \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{1}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \\[1em] \vdots & \ddots & \vdots \\[1em] \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{1}} & \dots & \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{n}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{m}} \end{bmatrix} $$
$$ F_{ij}^{\prime} = \sum \limits_{\ell=1}^{k} \dfrac{\partial f_{i}}{\partial g_{\ell}} \dfrac{\partial g_{\ell}}{\partial x_{j}},\quad 1\le i \le n, 1 \le j \le m $$
증명
일반화된 증명을 참고하자.
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