find the difference quotient $\frac{f(x + h)-f(x)}{h}$, where $h\neq0$, for the function below.\n\n$f(x)=2x^{…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=2(x + h)^2+2(x + h)\ &=2(x^{2}+2xh+h^{2})+2x + 2h\ &=2x^{2}+4xh+2h^{2}+2x + 2h \end{align*} ]

Step2: Calculate $f(x + h)-f(x)$

[ \begin{align*} f(x + h)-f(x)&=(2x^{2}+4xh+2h^{2}+2x + 2h)-(2x^{2}+2x)\ &=2x^{2}+4xh+2h^{2}+2x + 2h - 2x^{2}-2x\ &=4xh+2h^{2}+2h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{4xh+2h^{2}+2h}{h}\ &=\frac{h(4x + 2h+2)}{h}\ &=4x + 2h+2 \end{align*} ]

Answer:

$4x + 2h+2$