$$$\tan{\left(\frac{\theta}{2} \right)}$$$の導関数

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

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

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

正接関数の導関数は$$$\frac{d}{du} \left(\tan{\left(u \right)}\right) = \sec^{2}{\left(u \right)}$$$:

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

元の変数に戻す:

$$\sec^{2}{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(\frac{\theta}{2}\right) = \sec^{2}{\left({\color{red}\left(\frac{\theta}{2}\right)} \right)} \frac{d}{d\theta} \left(\frac{\theta}{2}\right)$$

定数倍の法則 $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ を $$$c = \frac{1}{2}$$$ と $$$f{\left(\theta \right)} = \theta$$$ に対して適用します:

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

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

$$\frac{\sec^{2}{\left(\frac{\theta}{2} \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)}}{2} = \frac{\sec^{2}{\left(\frac{\theta}{2} \right)} {\color{red}\left(1\right)}}{2}$$

簡単化せよ:

$$\frac{\sec^{2}{\left(\frac{\theta}{2} \right)}}{2} = \frac{1}{\cos{\left(\theta \right)} + 1}$$

したがって、$$$\frac{d}{d\theta} \left(\tan{\left(\frac{\theta}{2} \right)}\right) = \frac{1}{\cos{\left(\theta \right)} + 1}$$$。