Funktion $$$x^{3} - 3 x + 2$$$ derivaatta

Summan/erotuksen derivaatta on derivaattojen summa/erotus:

$${\color{red}\left(\frac{d}{dx} \left(x^{3} - 3 x + 2\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{3}\right) - \frac{d}{dx} \left(3 x\right) + \frac{d}{dx} \left(2\right)\right)}$$

Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = 3$$$ ja $$$f{\left(x \right)} = x$$$:

$$- {\color{red}\left(\frac{d}{dx} \left(3 x\right)\right)} + \frac{d}{dx} \left(2\right) + \frac{d}{dx} \left(x^{3}\right) = - {\color{red}\left(3 \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(2\right) + \frac{d}{dx} \left(x^{3}\right)$$

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

$$- 3 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(2\right) + \frac{d}{dx} \left(x^{3}\right) = - 3 {\color{red}\left(1\right)} + \frac{d}{dx} \left(2\right) + \frac{d}{dx} \left(x^{3}\right)$$

Vakion derivaatta on $$$0$$$:

$${\color{red}\left(\frac{d}{dx} \left(2\right)\right)} + \frac{d}{dx} \left(x^{3}\right) - 3 = {\color{red}\left(0\right)} + \frac{d}{dx} \left(x^{3}\right) - 3$$

Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$, kun $$$n = 3$$$:

$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} - 3 = {\color{red}\left(3 x^{2}\right)} - 3$$

Näin ollen, $$$\frac{d}{dx} \left(x^{3} - 3 x + 2\right) = 3 x^{2} - 3$$$.