Funktion $$$\sin{\left(a - x \right)}$$$ derivaatta muuttujan $$$x$$$ suhteen

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

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

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

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

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

Palaa alkuperäiseen muuttujaan:

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

Summan/erotuksen derivaatta on derivaattojen summa/erotus:

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

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

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

Vakion derivaatta on $$$0$$$:

$$\left({\color{red}\left(\frac{da}{dx}\right)} - 1\right) \cos{\left(a - x \right)} = \left({\color{red}\left(0\right)} - 1\right) \cos{\left(a - x \right)}$$

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