# How do you use the definition of a derivative to find the derivative of f(x)=3x+2?

Dec 7, 2016

$f ' \left(x\right) = \frac{\mathrm{df}}{\mathrm{dx}} = 3$
(see below for method using the definition of a derivative).

#### Explanation:

The definition of the derivative of $f \left(x\right)$ is
$\textcolor{w h i t e}{\text{XXX}} f ' \left(x\right) = \frac{\mathrm{df}}{\mathrm{dx}} = {\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$

For the example $f \left(x\right) = 3 x + 2$
$\textcolor{w h i t e}{\text{XXX}} \frac{\mathrm{df}}{\mathrm{dx}} = {\lim}_{h \rightarrow 0} \frac{\left(3 \left(x + h\right) + 2\right) - \left(3 x + 2\right)}{h}$

$\textcolor{w h i t e}{\text{XXXXX}} = {\lim}_{h r a r 0} \frac{\cancel{3 x} + 3 h \cancel{+ 2} \cancel{- 3 x} \cancel{- 2}}{h}$

$\textcolor{w h i t e}{\text{XXXXXX}} = {\lim}_{\leftrightarrow r 0} \frac{3 \cancel{h}}{\cancel{h}}$

$\textcolor{w h i t e}{\text{XXXXXX}} = {\lim}_{h \rightarrow 0} 3$

$\textcolor{w h i t e}{\text{XXXXXX}} = 3$