question 12 of 13, step 1 of 1 given the following function, determine the difference quotient, (f(x +…

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=6(x + h)^2+(x + h)-4\ &=6(x^{2}+2xh+h^{2})+x + h-4\ &=6x^{2}+12xh+6h^{2}+x + h-4 \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(6x^{2}+12xh+6h^{2}+x + h-4)-(6x^{2}+x - 4)\ &=6x^{2}+12xh+6h^{2}+x + h-4 - 6x^{2}-x + 4\ &=12xh+6h^{2}+h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{12xh+6h^{2}+h}{h}\ &=\frac{h(12x + 6h+1)}{h}\ &=12x+6h + 1 \end{align*} ]

Answer:

$12x+6h + 1$