Question

Find the difference quotient for f(x)=2x+1

Answer 1

The difference quotient or simply the derivative of a function is,

$\dfrac{d}{dx}f(x)=\lim_{\Delta x\to0} \dfrac{f(x+\Delta x)-f(x)}{\Delta x}$

Here the derivative is,

$\dfrac{d}{dx}2x+1=\lim_{\Delta x\to0}\dfrac{2(x+\Delta x)+1-(2x+1)}{\Delta x} \\ \implies\lim_{\Delta x\to 0}\dfrac{2\Delta x}{\Delta x}=2$

So the difference quotient of your linear function is its slope namely 2.

Hope this helps.