(related to Corollary: Derivative of a Linear Function \(ax+b\))
Given a linear function \(f(x)=ax +b\), \(a,b\in\mathbb R\) , and by definition is of the derivative we have
\[f'(x)=\lim_{\substack{\xi\to x\\\xi\neq x}}\frac {f(\xi)-f(x)}{\xi-x}=\frac{a\xi+b-ax-b}{\xi-x}=a\cdot\frac{\xi-x}{\xi-x}=a.\]
Note that the derivative of a linear function does not depend on the constant \(b\), because it cancels out.