Derivatan av $$$3 \sin{\left(x \right)} - 2$$$

Derivatan av en summa/differens är summan/differensen av derivatorna:

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

Derivatan av en konstant är $$$0$$$:

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

Tillämpa konstantfaktorregeln $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ med $$$c = 3$$$ och $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:

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

Derivatan av sinus är $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

$$3 {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = 3 {\color{red}\left(\cos{\left(x \right)}\right)}$$

Alltså, $$$\frac{d}{dx} \left(3 \sin{\left(x \right)} - 2\right) = 3 \cos{\left(x \right)}$$$.