Derivada de $$$1 - \sin{\left(\frac{t}{2} \right)}$$$

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

$${\color{red}\left(\frac{d}{dt} \left(1 - \sin{\left(\frac{t}{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{dt} \left(1\right) - \frac{d}{dt} \left(\sin{\left(\frac{t}{2} \right)}\right)\right)}$$

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

$${\color{red}\left(\frac{d}{dt} \left(1\right)\right)} - \frac{d}{dt} \left(\sin{\left(\frac{t}{2} \right)}\right) = {\color{red}\left(0\right)} - \frac{d}{dt} \left(\sin{\left(\frac{t}{2} \right)}\right)$$

La función $$$\sin{\left(\frac{t}{2} \right)}$$$ es la composición $$$f{\left(g{\left(t \right)} \right)}$$$ de dos funciones $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ y $$$g{\left(t \right)} = \frac{t}{2}$$$.

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

$$- {\color{red}\left(\frac{d}{dt} \left(\sin{\left(\frac{t}{2} \right)}\right)\right)} = - {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dt} \left(\frac{t}{2}\right)\right)}$$

La derivada del seno es $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$$- {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dt} \left(\frac{t}{2}\right) = - {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dt} \left(\frac{t}{2}\right)$$

Volver a la variable original:

$$- \cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dt} \left(\frac{t}{2}\right) = - \cos{\left({\color{red}\left(\frac{t}{2}\right)} \right)} \frac{d}{dt} \left(\frac{t}{2}\right)$$

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

$$- \cos{\left(\frac{t}{2} \right)} {\color{red}\left(\frac{d}{dt} \left(\frac{t}{2}\right)\right)} = - \cos{\left(\frac{t}{2} \right)} {\color{red}\left(\frac{\frac{d}{dt} \left(t\right)}{2}\right)}$$

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

$$- \frac{\cos{\left(\frac{t}{2} \right)} {\color{red}\left(\frac{d}{dt} \left(t\right)\right)}}{2} = - \frac{\cos{\left(\frac{t}{2} \right)} {\color{red}\left(1\right)}}{2}$$

Por lo tanto, $$$\frac{d}{dt} \left(1 - \sin{\left(\frac{t}{2} \right)}\right) = - \frac{\cos{\left(\frac{t}{2} \right)}}{2}$$$.