$$$x$$$ に関する $$$x^{2} y^{2} = 2 x + e^{y}$$$ の陰関数微分

方程式の両辺をそれぞれ微分せよ ($$$y$$$ を $$$x$$$ の関数として扱う): $$$\frac{d}{dx} \left(x^{2} y^{2}{\left(x \right)}\right) = \frac{d}{dx} \left(2 x + e^{y{\left(x \right)}}\right)$$$

方程式の左辺を微分せよ。

積の微分法 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を $$$f{\left(x \right)} = x^{2}$$$ と $$$g{\left(x \right)} = y^{2}{\left(x \right)}$$$ に適用する:

$${\color{red}\left(\frac{d}{dx} \left(x^{2} y^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) y^{2}{\left(x \right)} + x^{2} \frac{d}{dx} \left(y^{2}{\left(x \right)}\right)\right)}$$

関数$$$y^{2}{\left(x \right)}$$$は、2つの関数$$$f{\left(u \right)} = u^{2}$$$と$$$g{\left(x \right)} = y{\left(x \right)}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

$$x^{2} {\color{red}\left(\frac{d}{dx} \left(y^{2}{\left(x \right)}\right)\right)} + y^{2}{\left(x \right)} \frac{d}{dx} \left(x^{2}\right) = x^{2} {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(y{\left(x \right)}\right)\right)} + y^{2}{\left(x \right)} \frac{d}{dx} \left(x^{2}\right)$$

冪法則 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ を $$$n = 2$$$ に対して適用する:

$$x^{2} {\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + y^{2}{\left(x \right)} \frac{d}{dx} \left(x^{2}\right) = x^{2} {\color{red}\left(2 u\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + y^{2}{\left(x \right)} \frac{d}{dx} \left(x^{2}\right)$$

元の変数に戻す:

$$2 x^{2} {\color{red}\left(u\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + y^{2}{\left(x \right)} \frac{d}{dx} \left(x^{2}\right) = 2 x^{2} {\color{red}\left(y{\left(x \right)}\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + y^{2}{\left(x \right)} \frac{d}{dx} \left(x^{2}\right)$$

冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を $$$n = 2$$$ に対して適用する:

$$2 x^{2} y{\left(x \right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + y^{2}{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 2 x^{2} y{\left(x \right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + y^{2}{\left(x \right)} {\color{red}\left(2 x\right)}$$

簡単化せよ:

$$2 x^{2} y{\left(x \right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + 2 x y^{2}{\left(x \right)} = 2 x \left(x \frac{d}{dx} \left(y{\left(x \right)}\right) + y{\left(x \right)}\right) y{\left(x \right)}$$

したがって、$$$\frac{d}{dx} \left(x^{2} y^{2}{\left(x \right)}\right) = 2 x \left(x \frac{d}{dx} \left(y{\left(x \right)}\right) + y{\left(x \right)}\right) y{\left(x \right)}$$$。

方程式の右辺を微分する。

和/差の導関数は、導関数の和/差である:

$${\color{red}\left(\frac{d}{dx} \left(2 x + e^{y{\left(x \right)}}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(2 x\right) + \frac{d}{dx} \left(e^{y{\left(x \right)}}\right)\right)}$$

関数$$$e^{y{\left(x \right)}}$$$は、2つの関数$$$f{\left(u \right)} = e^{u}$$$と$$$g{\left(x \right)} = y{\left(x \right)}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。

連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:

$${\color{red}\left(\frac{d}{dx} \left(e^{y{\left(x \right)}}\right)\right)} + \frac{d}{dx} \left(2 x\right) = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(y{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(2 x\right)$$

指数関数の微分は$$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$です:

$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + \frac{d}{dx} \left(2 x\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(y{\left(x \right)}\right) + \frac{d}{dx} \left(2 x\right)$$

元の変数に戻す:

$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + \frac{d}{dx} \left(2 x\right) = e^{{\color{red}\left(y{\left(x \right)}\right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + \frac{d}{dx} \left(2 x\right)$$

定数倍の法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ を $$$c = 2$$$ と $$$f{\left(x \right)} = x$$$ に対して適用します:

$$e^{y{\left(x \right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + {\color{red}\left(\frac{d}{dx} \left(2 x\right)\right)} = e^{y{\left(x \right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + {\color{red}\left(2 \frac{d}{dx} \left(x\right)\right)}$$

$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$e^{y{\left(x \right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + 2 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = e^{y{\left(x \right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + 2 {\color{red}\left(1\right)}$$

したがって、$$$\frac{d}{dx} \left(2 x + e^{y{\left(x \right)}}\right) = e^{y{\left(x \right)}} \frac{d}{dx} \left(y{\left(x \right)}\right) + 2$$$。

したがって、導関数に関する次の線形方程式が得られた：$$$2 x y \left(x \frac{dy}{dx} + y\right) = e^{y} \frac{dy}{dx} + 2$$$。

これを解くと、$$$\frac{dy}{dx} = \frac{- 2 x y^{2} + 2}{2 x^{2} y - e^{y}}$$$ となる。