How do you use the definition of a derivative to find the derivative of  f(x) = 5x + 9 at x=2?

Oct 31, 2016

$f ' \left(2\right) = 5$

Explanation:

By definition $f ' \left(x\right) = {\lim}_{h \rightarrow 0} \left(\frac{f \left(x + h\right) - f \left(x\right)}{h}\right)$

So, with $f \left(x\right) = 5 x + 9$ we have:

$f ' \left(2\right) = {\lim}_{h \rightarrow 0} \left(\frac{f \left(2 + h\right) - f \left(2\right)}{h}\right)$
$\therefore f ' \left(2\right) = {\lim}_{h \rightarrow 0} \left(\frac{\left(5 \left(2 + h\right) + 9\right) - \left(5 \left(2\right) + 9\right)}{h}\right)$
$\therefore f ' \left(2\right) = {\lim}_{h \rightarrow 0} \left(\frac{\left(10 + 5 h + 9\right) - \left(10 + 9\right)}{h}\right)$
$\therefore f ' \left(2\right) = {\lim}_{h \rightarrow 0} \left(\frac{10 + 5 h + 9 - 10 - 9}{h}\right)$
$\therefore f ' \left(2\right) = {\lim}_{h \rightarrow 0} \left(\frac{5 h}{h}\right)$
$\therefore f ' \left(2\right) = {\lim}_{h \rightarrow 0} \left(5\right)$
$\therefore f ' \left(2\right) = 5$