The difference quotient or simply the derivative of a function is,
[tex]\dfrac{d}{dx}f(x)=\lim_{\Delta x\to0} \dfrac{f(x+\Delta x)-f(x)}{\Delta x}[/tex]
Here the derivative is,
[tex]\dfrac{d}{dx}2x+1=\lim_{\Delta x\to0}\dfrac{2(x+\Delta x)+1-(2x+1)}{\Delta x} \\ \implies\lim_{\Delta x\to 0}\dfrac{2\Delta x}{\Delta x}=2[/tex]
So the difference quotient of your linear function is its slope namely 2.
Hope this helps.
