Answer
a) The derivative of $f\left( x \right)={{x}^{3}}+2$ at x is $f'\left( x \right)=3{{x}^{2}}$.
b) The slope of the tangent line to the graph of $f\left( x \right)={{x}^{3}}+2$ at $x=-1$ is $f'\left( -1 \right)=3$ and at $x=1$ is $f'\left( 1 \right)=3$.
Work Step by Step
(a)
Consider the function, $f\left( x \right)={{x}^{3}}+2$
Now, compute the derivate of $f\left( x \right)={{x}^{3}}+2$ using the formula $f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( x+h \right)-f\left( x \right)}{h}$
To compute $f\left( x+h \right)$, substitute $x=x+h$ in the function $f\left( x \right)={{x}^{3}}+2$.
$f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{\left[ {{\left( x+h \right)}^{3}}+2 \right]-\left( {{x}^{3}}+2 \right)}{h}$
Now, simplify ${{\left( x+h \right)}^{2}}$ by using the property ${{\left( A+B \right)}^{3}}={{A}^{3}}+3A{{B}^{2}}+3{{A}^{2}}B+{{B}^{3}}$.
$f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{{{x}^{3}}+3h{{x}^{2}}+3{{h}^{2}}x+{{h}^{3}}+2-{{x}^{3}}-2}{h}$
Combine the like terms in the numerator; this gives,
$f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{3h{{x}^{2}}+3{{h}^{2}}x+{{h}^{3}}}{h}$
Divide numerator and denominator by h.
$f'\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\left( 3{{x}^{2}}+3hx+{{h}^{2}} \right)$
Apply the limits,
$\begin{align}
& f'\left( x \right)=3{{x}^{2}}+3hx+{{h}^{2}} \\
& =3{{x}^{2}}+3\cdot 0\cdot x+{{0}^{2}} \\
& =3{{x}^{2}}
\end{align}$
Thus, the derivative of $f\left( x \right)={{x}^{3}}+2$ at x is $f'\left( x \right)=3{{x}^{2}}$.
(b)
Consider the function, $f\left( x \right)={{x}^{3}}+2$
From part (a), the derivative of $f\left( x \right)={{x}^{3}}+2$ at x is $f'\left( x \right)=3{{x}^{2}}$.
To compute the slope of the tangent line at $x=-1$, substitute $x=-1$ in $f'\left( x \right)$.
$\begin{align}
& f'\left( -1 \right)=3{{\left( -1 \right)}^{2}} \\
& =3\cdot 1 \\
& =3
\end{align}$
To compute the slope of the tangent line at $x=1$, substitute $x=1$ in $f'\left( x \right)$.
$\begin{align}
& f'\left( 1 \right)=3{{\left( 1 \right)}^{2}} \\
& =3\cdot 1 \\
& =3
\end{align}$
