$$$\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$ 的導數

函數 $$$\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$$ 是兩個函數 $$$f{\left(u \right)} = \tan{\left(u \right)}$$$ 與 $$$g{\left(x \right)} = \frac{x}{2} + \frac{\pi}{4}$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。

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

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

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

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

返回原變數：

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

和/差的導數等於導數的和/差：

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

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 $$$c = \frac{1}{2}$$$ 與 $$$f{\left(x \right)} = x$$$：

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

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

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

常數的導數為$$$0$$$：

$$\left({\color{red}\left(\frac{d}{dx} \left(\frac{\pi}{4}\right)\right)} + \frac{1}{2}\right) \sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)} = \left({\color{red}\left(0\right)} + \frac{1}{2}\right) \sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}$$

化簡：

$$\frac{\sec^{2}{\left(\frac{x}{2} + \frac{\pi}{4} \right)}}{2} = \frac{1}{1 - \sin{\left(x \right)}}$$

因此，$$$\frac{d}{dx} \left(\tan{\left(\frac{x}{2} + \frac{\pi}{4} \right)}\right) = \frac{1}{1 - \sin{\left(x \right)}}$$$。