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

A derivada de uma soma/diferença é a soma/diferença das derivadas:

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

A derivada de uma constante é $$$0$$$:

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

A função $$$\sin{\left(\frac{t}{2} \right)}$$$ é a composição $$$f{\left(g{\left(t \right)} \right)}$$$ de duas funções $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ e $$$g{\left(t \right)} = \frac{t}{2}$$$.

Aplique a regra da cadeia $$$\frac{d}{dt} \left(f{\left(g{\left(t \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dt} \left(g{\left(t \right)}\right)$$$:

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

A derivada do seno é $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

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

Retorne à variável original:

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

Aplique a regra da constante multiplicativa $$$\frac{d}{dt} \left(c f{\left(t \right)}\right) = c \frac{d}{dt} \left(f{\left(t \right)}\right)$$$ com $$$c = \frac{1}{2}$$$ e $$$f{\left(t \right)} = t$$$:

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

Aplique a regra da potência $$$\frac{d}{dt} \left(t^{n}\right) = n t^{n - 1}$$$ com $$$n = 1$$$, em outras palavras, $$$\frac{d}{dt} \left(t\right) = 1$$$:

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

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