$$$\sin{\left(x^{2} \right)}$$$ の二階導関数

一階導関数 $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)$$$ を求めよ

関数$$$\sin{\left(x^{2} \right)}$$$は、2つの関数$$$f{\left(u \right)} = \sin{\left(u \right)}$$$と$$$g{\left(x \right)} = x^{2}$$$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:

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

正弦関数の導関数は$$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

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

元の変数に戻す:

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

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:

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

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

次に、$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = \frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right)$$$

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 $$$f{\left(x \right)} = x \cos{\left(x^{2} \right)}$$$ に対して適用します:

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

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 $$$f{\left(x \right)} = x$$$ と $$$g{\left(x \right)} = \cos{\left(x^{2} \right)}$$$ に適用する:

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

関数$$$\cos{\left(x^{2} \right)}$$$は、2つの関数$$$f{\left(u \right)} = \cos{\left(u \right)}$$$と$$$g{\left(x \right)} = x^{2}$$$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:

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

余弦関数の導関数は$$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:

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

元の変数に戻す:

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

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:

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

$$$n = 1$$$ 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 $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$- 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)} {\color{red}\left(1\right)}$$

したがって、$$$\frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$。

したがって、$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$。