find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=3x² + 3x simplify…

find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=3x² + 3x simplify your answer as much as possible. (f(x + h)-f(x))/h =

find the difference quotient (f(x + h)-f(x))/h, where h≠0, for the function below. f(x)=3x² + 3x simplify your answer as much as possible. (f(x + h)-f(x))/h =

Answer

Explanation:

Step1: Find $f(x + h)$

Substitute $x+h$ into $f(x)$: [ \begin{align*} f(x + h)&=3(x + h)^2+3(x + h)\ &=3(x^{2}+2xh+h^{2})+3x + 3h\ &=3x^{2}+6xh+3h^{2}+3x + 3h \end{align*} ]

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

[ \begin{align*} f(x + h)-f(x)&=(3x^{2}+6xh+3h^{2}+3x + 3h)-(3x^{2}+3x)\ &=3x^{2}+6xh+3h^{2}+3x + 3h - 3x^{2}-3x\ &=6xh+3h^{2}+3h \end{align*} ]

Step3: Find the difference - quotient

[ \begin{align*} \frac{f(x + h)-f(x)}{h}&=\frac{6xh+3h^{2}+3h}{h}\ &=\frac{h(6x + 3h+3)}{h}\ &=6x+3h + 3 \end{align*} ]

Answer:

$6x+3h + 3$