Derivada de $$$\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}$$$

La función $$$\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}$$$ es la composición $$$f{\left(g{\left(\theta \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \tan{\left(u \right)}$$$ y $$$g{\left(\theta \right)} = \frac{\theta}{2} + \frac{\pi}{4}$$$.

Aplica la regla de la cadena $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$:

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

La derivada de la tangente es $$$\frac{d}{du} \left(\tan{\left(u \right)}\right) = \sec^{2}{\left(u \right)}$$$:

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

Volver a la variable original:

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

La derivada de una suma/diferencia es la suma/diferencia de las derivadas:

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

Aplica la regla del factor constante $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$ con $$$c = \frac{1}{2}$$$ y $$$f{\left(\theta \right)} = \theta$$$:

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

Aplica la regla de la potencia $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$ con $$$n = 1$$$, en otras palabras, $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:

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

La derivada de una constante es $$$0$$$:

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

Simplificar:

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

Por lo tanto, $$$\frac{d}{d\theta} \left(\tan{\left(\frac{\theta}{2} + \frac{\pi}{4} \right)}\right) = \frac{1}{1 - \sin{\left(\theta \right)}}$$$.