consider the function below.
$f(x)=5x^{3}+6x$
find the difference quotient below (where $h
eq0$) and simplify your answer.
$\frac{f(x + h)-f(x)}{h}$

Question

Accepted Answer

# Explanation:
## Step1: Find $f(x + h)$
Substitute $x+h$ into $f(x)$:
$$
\begin{align*}
f(x + h)&=5(x + h)^{3}+6(x + h)\
&=5(x^{3}+3x^{2}h + 3xh^{2}+h^{3})+6x+6h\
&=5x^{3}+15x^{2}h+15xh^{2}+5h^{3}+6x + 6h
\end{align*}
$$
## Step2: Calculate $f(x + h)-f(x)$
$$
\begin{align*}
f(x + h)-f(x)&=(5x^{3}+15x^{2}h+15xh^{2}+5h^{3}+6x + 6h)-(5x^{3}+6x)\
&=5x^{3}+15x^{2}h+15xh^{2}+5h^{3}+6x + 6h - 5x^{3}-6x\
&=15x^{2}h+15xh^{2}+5h^{3}+6h
\end{align*}
$$
## Step3: Find the difference - quotient $\frac{f(x + h)-f(x)}{h}$
$$
\begin{align*}
\frac{f(x + h)-f(x)}{h}&=\frac{15x^{2}h+15xh^{2}+5h^{3}+6h}{h}\
&=\frac{h(15x^{2}+15xh + 5h^{2}+6)}{h}\
&=15x^{2}+15xh+5h^{2}+6
\end{align*}
$$
# Answer:
$15x^{2}+15xh + 5h^{2}+6$

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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 \(\frac{f(x + h)-f(x)}{h}\)

Answer: