Question

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

Answer 1

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

`(d)/(dx)f(x)=\lim_(\Delta x\to0) (f(x+\Delta x)-f(x))/(\Delta x)`

Here the derivative is,

`(d)/(dx)2x+1=\lim_(\Delta x\to0)(2(x+\Delta x)+1-(2x+1))/(\Delta x) \\ \implies\lim_(\Delta x\to 0)(2\Delta x)/(\Delta x)=2`

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

Hope this helps.