Calculus and Its Applications, Chapter 1, 1.3, Section 1.3, Problem 50

Determine the simplified difference quotient of the function $f(x) = x^3 + x$
For $f(x) = x^3 + x$

$
\begin{equation}
\begin{aligned}
f(x + h) &= (x + h)^3 + (x + h)\\
\\
&= x^3 + 3xh + 3xh^2 + h^3 + x + h
\end{aligned}
\end{equation}
$

Then,

$
\begin{equation}
\begin{aligned}
f(x +h) - f(x) &= x^3 + 3xh + 3xh^2 + h^3 + x + h - (x^3 + x)\\
\\
&= x^3 + 3xh + 3xh^2 + h^3 + x + h -x^3 -x \\
\\
&= 3x^2h + 3xh^2 + h^3 + h
\end{aligned}
\end{equation}
$


Thus,

$
\begin{equation}
\begin{aligned}
\frac{f(x+h)-f(x)}{h} &= \frac{3x^2h + 3xh^2 + h^3 + h}{h}\\
\\
&= \frac{h\left( 3x^2 + 3xh + h^2 + 1 \right)}{h}\\
\\
&= 3x^2 + 3xh + h^2 + 1
\end{aligned}
\end{equation}
$