$$$\tan^{2}{\left(x \right)}$$$ 的二階導數

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

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

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

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

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

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

返回原變數：

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

正切函數的導數為 $$$\frac{d}{dx} \left(\tan{\left(x \right)}\right) = \sec^{2}{\left(x \right)}$$$：

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

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

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

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 $$$f{\left(x \right)} = \tan{\left(x \right)} \sec^{2}{\left(x \right)}$$$：

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

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 $$$f{\left(x \right)} = \sec^{2}{\left(x \right)}$$$ 和 $$$g{\left(x \right)} = \tan{\left(x \right)}$$$：

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

正切函數的導數為 $$$\frac{d}{dx} \left(\tan{\left(x \right)}\right) = \sec^{2}{\left(x \right)}$$$：

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

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

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

$$2 \tan{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\sec^{2}{\left(x \right)}\right)\right)} + 2 \sec^{4}{\left(x \right)} = 2 \tan{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\sec{\left(x \right)}\right)\right)} + 2 \sec^{4}{\left(x \right)}$$

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

$$2 \tan{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right) + 2 \sec^{4}{\left(x \right)} = 2 \tan{\left(x \right)} {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right) + 2 \sec^{4}{\left(x \right)}$$

返回原變數：

$$4 \tan{\left(x \right)} {\color{red}\left(u\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right) + 2 \sec^{4}{\left(x \right)} = 4 \tan{\left(x \right)} {\color{red}\left(\sec{\left(x \right)}\right)} \frac{d}{dx} \left(\sec{\left(x \right)}\right) + 2 \sec^{4}{\left(x \right)}$$

正割函數的導數為 $$$\frac{d}{dx} \left(\sec{\left(x \right)}\right) = \tan{\left(x \right)} \sec{\left(x \right)}$$$：

$$4 \tan{\left(x \right)} \sec{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\sec{\left(x \right)}\right)\right)} + 2 \sec^{4}{\left(x \right)} = 4 \tan{\left(x \right)} \sec{\left(x \right)} {\color{red}\left(\tan{\left(x \right)} \sec{\left(x \right)}\right)} + 2 \sec^{4}{\left(x \right)}$$

化簡：

$$4 \tan^{2}{\left(x \right)} \sec^{2}{\left(x \right)} + 2 \sec^{4}{\left(x \right)} = \left(-4 + \frac{6}{\cos^{2}{\left(x \right)}}\right) \sec^{2}{\left(x \right)}$$

因此，$$$\frac{d}{dx} \left(2 \tan{\left(x \right)} \sec^{2}{\left(x \right)}\right) = \left(-4 + \frac{6}{\cos^{2}{\left(x \right)}}\right) \sec^{2}{\left(x \right)}$$$。

因此，$$$\frac{d^{2}}{dx^{2}} \left(\tan^{2}{\left(x \right)}\right) = \left(-4 + \frac{6}{\cos^{2}{\left(x \right)}}\right) \sec^{2}{\left(x \right)}$$$。