172. Find the derivative of the function $y=x^{2}+x$.
Solution. $1^{0} . y_{\text {n }}=(x+\Delta x)^{2}+(x+\Delta x)=x^{2}+2 x \Delta x+(\Delta x)^{2}+x+\Delta x$.

$2^{0} . \Delta y=y_{\mathrm{N}}-y=\left(x^{2}+2 x \Delta x+(\Delta x)^{2}+x+\Delta x\right)-\left(x^{2}+x\right)=x^{2}+2 x \Delta x+$ $+(\Delta x)^{2}+x+\Delta x-x^{2}-x=2 x \Delta x+(\Delta x)^{2}+\Delta x$.

$3^{0} \cdot \frac{\Delta y}{\Delta x}=\frac{2 x \Delta x+(\Delta x)^{2}+\Delta x}{\Delta x}=2 x+\Delta x+1$.