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+6\ &=2(x^{2}+2xh+h^{2})+6\ &=2x^{2}+4xh + 2h^{2}+6 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(2x^{2}+4xh + 2h^{2}+6)-(2x^{2}+6)\ &=2x^{2}+4xh + 2h^{2}+6 - 2x^{2}-6\ &=4xh+2h^{2} \end{align*} ]

Step3: Find the difference quotient

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

Answer:

$4x + 2h$