Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 17

Find the derivative of $\displaystyle f(x) = \frac{1}{2}x - \frac{1}{3}$ using the definition and the domain of its derivative.

Using the definition of derivative


$
\begin{equation}
\begin{aligned}

\qquad f'(x) =& \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
&&
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\displaystyle \frac{x + h }{2} - \frac{1}{3} - \left( \frac{x}{2} - \frac{1}{3}\right)}{h}
&& \text{Substitute $f(x + h)$ and $f(x)$}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\displaystyle \cancel{\frac{x}{2}} + \frac{h}{2} - \cancel{\frac{1}{3}} - \cancel{\frac{x}{2}} + \cancel{\frac{1}{3}}}{h}
&& \text{Combine like terms}
\\
\\
\qquad f'(x) =& \lim_{h \to 0} \frac{\cancel{h}}{2\cancel{h}}
&& \text{Cancel out like terms}

\end{aligned}
\end{equation}
$


$\qquad \fbox{$ f'(x) = \displaystyle \frac{1}{2}$}$

Both $f(x)$ and $f'(x)$ are linear functions that extend in every number. Therefore, their domain is $(-\infty, \infty)$