$$$\sin{\left(x^{2} \right)}$$$ 的二階導數

求第一階導數 $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)$$$

函數 $$$\sin{\left(x^{2} \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 與 $$$g{\left(x \right)} = x^{2}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。

應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：

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

正弦函數的導數為$$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$：

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

返回原變數：

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

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 2$$$：

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

因此，$$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right) = 2 x \cos{\left(x^{2} \right)}$$$。

接下來，$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = \frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right)$$$

套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = 2$$$ 與 $$$f{\left(x \right)} = x \cos{\left(x^{2} \right)}$$$：

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

將乘積法則 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ 應用於 $$$f{\left(x \right)} = x$$$ 和 $$$g{\left(x \right)} = \cos{\left(x^{2} \right)}$$$：

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

函數 $$$\cos{\left(x^{2} \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 與 $$$g{\left(x \right)} = x^{2}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。

應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：

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

餘弦函數的導數為 $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$：

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

返回原變數：

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

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 2$$$：

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

套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：

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

因此，$$$\frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$。

因此，$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$。