Question

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

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:

\(15x^{2}+15xh + 5h^{2}+6\)