The Power Rule and Linearity

The graph of a constant is a horizontal line, with slope $0$ everywhere. For a whole number $n \ge 2$, the power rule comes from the factoring identity

Divide by $x - a$ and let $x \to a$. Each of the $n$ terms in the second factor approaches $a^{n-1}$, so the derivative of $x^n$ at $a$ is $na^{n-1}$. For $n = 1$, the line $y = x$ has slope $1 = 1 \cdot x^0$.

The rule also covers roots and reciprocals, once they're written as powers. For $\sqrt[5]{x} = x^{1/5}$ it gives $\frac{1}{5}x^{-4/5}$, which is undefined at $x = 0$. The graph of $\sqrt[5]{x}$ has a vertical tangent there.

Both follow from the limit laws. The difference quotient of $cf$ is $c$ times the difference quotient of $f$, and the quotient of $f + g$ is the sum of their quotients. Together with the power rule, they differentiate every polynomial term by term.