The constant rule is the simplest and most easily understood rule. The derivative calculates the slope, right? So, if you are given a horizontal line, what is the slope? Right! The slope is zero. That's it. That's the slope of every horizontal line. We can write the equation of a horizontal line as \(f(x)=c\) where \(c\) is a real number. Since these are always horizontal lines, the slope is zero. Therefore, the derivative of all constant functions (horizontal lines) is zero. We can derive this idea from the limit definition as follows. If \(f(x)=c\) \[ f~'(x) = \lim_{h \to 0}{\frac{f(x+h) - f(x)}{h}} = \lim_{h \to 0}{\frac{c - c}{h}} = \lim_{h \to 0}{0} = 0 \] Notice that c is gone from the final answer, \(f~'(x)=0\), so this holds for all horizontal lines. Thinking about this, it makes sense intuitively, right?

This rule works as you would expect. Mathematically, it looks like this. \[ \frac{d}{dx}[cf(x)] = c \frac{d}{dx}[f(x)] \] Nothing surprising, just pull out the constant and take the derivative of the function. This is discussed in more detail with examples on the power rule page.

When you have two functions that are added or subtracted, you just take the derivative of each individually. Mathematically, it looks like this. \[ \frac{d}{dx}[f(x) \pm g(x)] = \frac{d}{dx}[f(x)] \pm \frac{d}{dx}[g(x)] \] Nothing surprising or tricky here. It works just as you would expect.
[However, you will find out soon that this idea does NOT hold for multiplication and division. We have some special rules for those called the product rule and quotient rule.]