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These rules you have nothing to worry about. They're there to make differentiation easier to do, that's all.

Examples:

#1,2 Let's say you have:

\(f\left(x\right)=x±x^2\)

You differentiate twice:

\(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(x^2\right)\)

If they're being subtracted (minus), it's:

\(\frac{d}{dx}\left(x\right)-\frac{d}{dx}\left(x^2\right)\)

#3 Let's say you have:

\(f\left(x\right)=3x^2\)

This becomes:

\(3\left[\frac{d}{dx}\left(x^2\right)\right]\)

#4 The best rule! Anytime your teacher asks you to differentiate a number without any algebra, it's 0 automatically. For example:

\(f\left(x\right)=3x^2+5\) becomes:

\(\frac{d}{dx}3x^2+\frac{d}{dx}5\)

\(3\left[\frac{d}{dx}x^2\right]+0\) ...