Derivative of $$$\sin{\left(\frac{x}{2} - 1 \right)}$$$

The function $$$\sin{\left(\frac{x}{2} - 1 \right)}$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ and $$$g{\left(x \right)} = \frac{x}{2} - 1$$$.

Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(\frac{x}{2} - 1 \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(\frac{x}{2} - 1\right)\right)}$$

The derivative of the sine is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{x}{2} - 1\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{x}{2} - 1\right)$$

Return to the old variable:

$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{x}{2} - 1\right) = \cos{\left({\color{red}\left(\frac{x}{2} - 1\right)} \right)} \frac{d}{dx} \left(\frac{x}{2} - 1\right)$$

The derivative of a sum/difference is the sum/difference of derivatives:

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

The derivative of a constant is $$$0$$$:

$$\left(- {\color{red}\left(\frac{d}{dx} \left(1\right)\right)} + \frac{d}{dx} \left(\frac{x}{2}\right)\right) \cos{\left(\frac{x}{2} - 1 \right)} = \left(- {\color{red}\left(0\right)} + \frac{d}{dx} \left(\frac{x}{2}\right)\right) \cos{\left(\frac{x}{2} - 1 \right)}$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = \frac{1}{2}$$$ and $$$f{\left(x \right)} = x$$$:

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

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

Thus, $$$\frac{d}{dx} \left(\sin{\left(\frac{x}{2} - 1 \right)}\right) = \frac{\cos{\left(\frac{x}{2} - 1 \right)}}{2}$$$.