Answer
$\lim\limits_{\Delta x\to0^+}\dfrac{(x+\Delta x)^2+x+\Delta x-(x^2+x)}{\Delta x}=2x+1.$
Work Step by Step
$\lim\limits_{\Delta x\to0^+}\dfrac{(x+\Delta x)^2+x+\Delta x-(x^2+x)}{\Delta x}$
$=\lim\limits_{\Delta x\to0^+}\dfrac{x^2+2x(\Delta x)+(\Delta x)^2+x+\Delta x-x^2-x}{\Delta x}$
$=\lim\limits_{\Delta x\to0^+}\dfrac{\Delta x(2x+\Delta x+1)}{\Delta x}$
$=\lim\limits_{\Delta x\to0^+}2x+\Delta x+1=2x+0^++1=2x+1.$
