find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify it. f(x)=9x^2 + 3x the difference quotient is (simplify your answer. do not factor.)

Question

find the difference quotient $\frac{f(x + h)-f(x)}{h}$ for the function and simplify it. f(x)=9x^2 + 3x the difference quotient is  (simplify your answer. do not factor.)

Accepted Answer

# Explanation:
## Step1: Find f(x + h)
Substitute $x+h$ into $f(x)$:
$$
\begin{align*}
f(x + h)&=9(x + h)^2+3(x + h)\
&=9(x^{2}+2xh+h^{2})+3x + 3h\
&=9x^{2}+18xh+9h^{2}+3x + 3h
\end{align*}
$$
## Step2: Calculate f(x + h)-f(x)
$$
\begin{align*}
f(x + h)-f(x)&=(9x^{2}+18xh+9h^{2}+3x + 3h)-(9x^{2}+3x)\
&=9x^{2}+18xh+9h^{2}+3x + 3h - 9x^{2}-3x\
&=18xh+9h^{2}+3h
\end{align*}
$$
## Step3: Find the difference quotient
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{18xh+9h^{2}+3h}{h}\
&=\frac{h(18x + 9h+3)}{h}\
&=18x+9h + 3
\end{align*}
$$
# Answer:
$18x + 9h+3$

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QUESTION IMAGE

Question

Explanation:

Step1: Find f(x + h)

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

Step3: Find the difference quotient

Answer: