find the difference quotient of f; that is, find $\frac{f(x + h)-f(x)}{h}$, $h\neq0$, for the following…

Explanation:

Step1: Find f(x + h)

Substitute x + h into f(x): [ \begin{align*} f(x + h)&=2(x + h)^2-(x + h)-1\ &=2(x^{2}+2xh+h^{2})-x - h-1\ &=2x^{2}+4xh+2h^{2}-x - h-1 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(2x^{2}+4xh+2h^{2}-x - h-1)-(2x^{2}-x - 1)\ &=2x^{2}+4xh+2h^{2}-x - h-1 - 2x^{2}+x + 1\ &=4xh+2h^{2}-h \end{align*} ]

Step3: Find the difference quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{4xh+2h^{2}-h}{h}\ &=\frac{h(4x + 2h-1)}{h}\ &=4x+2h - 1 \end{align*} ]

Answer:

$4x + 2h-1$