Single Variable Calculus, Chapter 3, 3.2, Section 3.2, Problem 18

Find the derivative of $\displaystyle f(x) = mx + b$ using the definition and the domain of its 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{m (x + h) + b - (mx + b)}{h}
&& \text{Substitute $f(x + h)$ and $f(x)$}
\\
\\
\qquad f'(x) &= \lim_{h \to 0} \frac{\cancel{mx} + mh + \cancel{b} - \cancel{mx} - \cancel{b}}{h}
&& \text{Combine like terms}
\\
\\
\qquad f'(x) &= \lim_{h \to 0} \frac{m \cancel{h}}{\cancel{h}}
&& \text{Cancel out like terms}

\end{aligned}
\end{equation}
$


$\qquad \fbox{$f'(x) = m$}$

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

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