Funktion $$$\sin{\left(\frac{x}{2} - 1 \right)}$$$ derivaatta

Funktio $$$\sin{\left(\frac{x}{2} - 1 \right)}$$$ on kahden funktion $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ ja $$$g{\left(x \right)} = \frac{x}{2} - 1$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.

Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:

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

Sinin derivaatta on $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

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

Palaa alkuperäiseen muuttujaan:

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

Summan/erotuksen derivaatta on derivaattojen summa/erotus:

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

Vakion derivaatta on $$$0$$$:

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

Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = \frac{1}{2}$$$ ja $$$f{\left(x \right)} = x$$$:

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

Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ käyttäen $$$n = 1$$$, toisin sanoen, $$$\frac{d}{dx} \left(x\right) = 1$$$:

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

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