二階偏微分の連鎖律（chain rule）

更新 2025/08/07

定理

2変数関数 $f(u,v)$ について， $u = u(x,y), v = v(x,y)$ により $x,y$ 変数に置換したとき，2階偏微分は次のようになる。

$\begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial^2 f}{\partial u^2} \left( \dfrac{\partial u}{\partial x} \right)^2 + 2 \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial x} \dfrac{\partial v}{\partial x} + \dfrac{\partial^2 f}{\partial v^2} \left( \dfrac{\partial v}{\partial x} \right)^2\\ &\quad + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial x^2} \end{aligned}$

$\begin{aligned} \dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial^2 f}{\partial u^2} \dfrac{\partial u}{\partial x} \dfrac{\partial u}{\partial y} + \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial y} \dfrac{\partial v}{\partial x} + \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial x} \dfrac{\partial v}{\partial y}\\ &\quad + \dfrac{\partial^2 f}{\partial v^2} \dfrac{\partial v}{\partial x} \dfrac{\partial v}{\partial y} + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial x \partial y} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial x \partial y} \end{aligned}$

$\begin{aligned} \dfrac{\partial^2 f}{\partial y^2} &= \dfrac{\partial^2 f}{\partial u^2} \left( \dfrac{\partial u}{\partial y} \right)^2 + 2 \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial y} \dfrac{\partial v}{\partial y} + \dfrac{\partial^2 f}{\partial v^2} \left( \dfrac{\partial v}{\partial y} \right)^2\\ &\quad + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial y^2} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial y^2} \end{aligned}$

なお， $f,u,v$ はどれも $C^2$ 級関数としている。

この記事では2変数の合成関数の2階偏微分公式を証明します。

連鎖律（チェーンルール）を使い倒します。復習するときはこちらからどうぞ。→ 連鎖律（チェインルール）～多変数関数の合成関数の微分

証明

気合で計算します

$\dfrac{\partial^2 f}{\partial x^2}$

$\begin{aligned} \dfrac{\partial^2 f}{\partial x^2} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial u} \dfrac{\partial u}{\partial x} + \dfrac{\partial f}{\partial v} \dfrac{\partial v}{\partial x} \right) \end{aligned}$

積の微分公式より $\begin{aligned} (\text{式}) &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial u} \right) \dfrac{\partial u}{\partial x} + \dfrac{\partial f}{\partial u} \dfrac{\partial}{\partial x} \left( \dfrac{\partial u}{\partial x} \right) \\ &\quad + \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial v} \right) \dfrac{\partial v}{\partial x} + \dfrac{\partial f}{\partial v} \dfrac{\partial}{\partial x} \left( \dfrac{\partial v}{\partial x} \right)\\ \end{aligned}$ である。

チェーンルールをもう一度用いて整理すると $\begin{aligned} (\text{式}) &= \left( \dfrac{\partial^2 f}{\partial u^2} \dfrac{\partial u}{\partial x} + \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial v}{\partial x} \right) \dfrac{\partial u}{\partial x} + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial x^2} \\ &\quad + \left( \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial x} + \dfrac{\partial^2 f}{\partial^2 v} \dfrac{\partial v}{\partial x} \right) \dfrac{\partial v}{\partial x} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial x^2} \\ &= \dfrac{\partial^2 f}{\partial u^2} \left( \dfrac{\partial u}{\partial x} \right)^2 + 2 \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial x} \dfrac{\partial v}{\partial x} + \dfrac{\partial^2 f}{\partial v^2} \left( \dfrac{\partial v}{\partial x} \right)^2\\ &\quad + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial x^2} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial x^2} \end{aligned}$ となる。

$\dfrac{\partial^2 f}{\partial x \partial y}$

同様に計算する。 $\begin{aligned} \dfrac{\partial^2 f}{\partial x \partial y} &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial u} \dfrac{\partial u}{\partial y} + \dfrac{\partial f}{\partial v} \dfrac{\partial v}{\partial y} \right) \\ \end{aligned}$

積の微分公式を用いると $\begin{aligned} (\text{式}) &= \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial u} \right) \dfrac{\partial u}{\partial y} + \dfrac{\partial f}{\partial u} \dfrac{\partial}{\partial x} \left( \dfrac{\partial u}{\partial y} \right) \\ &\quad + \dfrac{\partial}{\partial x} \left( \dfrac{\partial f}{\partial v} \right) \dfrac{\partial v}{\partial y} + \dfrac{\partial f}{\partial v} \dfrac{\partial}{\partial x} \left( \dfrac{\partial v}{\partial y} \right)\\ \end{aligned}$ である。

チェーンルールをもう一度用いて整理すると $\begin{aligned} (\text{式}) &= \left( \dfrac{\partial^2 f}{\partial u^2} \dfrac{\partial u}{\partial x} + \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial v}{\partial x} \right) \dfrac{\partial u}{\partial y} + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial x \partial y} \\ &\quad + \left( \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial x} + \dfrac{\partial^2 f}{\partial^2 v} \dfrac{\partial v}{\partial x} \right) \dfrac{\partial v}{\partial y} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial x \partial y} \\ &= \dfrac{\partial^2 f}{\partial u^2} \dfrac{\partial u}{\partial x} \dfrac{\partial u}{\partial y} + \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial y} \dfrac{\partial v}{\partial x} + \dfrac{\partial^2 f}{\partial u \partial v} \dfrac{\partial u}{\partial x} \dfrac{\partial v}{\partial y}\\ &\quad + \dfrac{\partial^2 f}{\partial v^2} \dfrac{\partial v}{\partial x} \dfrac{\partial v}{\partial y} + \dfrac{\partial f}{\partial u} \dfrac{\partial^2 u}{\partial x \partial y} + \dfrac{\partial f}{\partial v} \dfrac{\partial^2 v}{\partial x \partial y} \end{aligned}$ となる。

$\dfrac{\partial^2 f}{\partial y^2}$

1番と同じなので省略。

計算上の注意点

$\dfrac{\partial u}{\partial x} \dfrac{\partial u}{\partial y} \neq \dfrac{\partial^2 u}{\partial x \partial y}$ です。しばしばこのように間違えてしまうことで，計算をミスしてしまう人がいます。

計算がたいへんですね。