Answer
The value of $\frac{f\left( x+h \right)-f\left( x \right)}{h}$ is $3{{x}^{2}}+3xh+{{h}^{2}}$.
Work Step by Step
Consider the function $ f\left( x \right)={{x}^{3}}$,
Find the value of $\frac{f\left( x+h \right)-f\left( x \right)}{h}$ using the definition of the function.
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{{{\left( x+h \right)}^{3}}-{{x}^{3}}}{h} \\
& =\frac{{{x}^{3}}+{{h}^{3}}+3xh\left( x+h \right)-{{x}^{3}}}{h} \\
& =\frac{{{h}^{3}}+3xh\left( x+h \right)}{h}
\end{align}$
Factoring out h from the numerator,
$\begin{align}
& \frac{f\left( x+h \right)-f\left( x \right)}{h}=\frac{h\left( {{h}^{2}}+3x\left( x+h \right) \right)}{h} \\
& ={{h}^{2}}+3x\left( x+h \right) \\
& ={{h}^{2}}+3{{x}^{2}}+3xh
\end{align}$
Thus, the value of $\frac{f\left( x+h \right)-f\left( x \right)}{h}$ is $3{{x}^{2}}+3xh+{{h}^{2}}$.
