Derivative/R/Affine-linear function/Directly/Example

Let ${}s,c\in \mathbb {R}$, and let

$\alpha \colon \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto sx+c,$

be an affine-linear function. To determine the derivative in a point ${}a\in \mathbb {R}$, we consider the difference quotient

${}{\frac {(sx+c)-(sa+c)}{x-a}}={\frac {sx-sa}{x-a}}=s\,.$

This is constant and equals ${}s$, so that the limit of the difference quotient as ${}x$ tends to ${}a$ exists and equals ${}s$ as well. Hence, the derivative exists in every point and is just ${}s$. The slope of the affine-linear function is also its derivative.