# How do you use the definition of a derivative to find the derivative of f(x)=1/3x+4/5?

Feb 6, 2017

$f ' \left(x\right) = \frac{1}{3}$

#### Explanation:

Using the $\textcolor{b l u e}{\text{limit definition}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{f ' \left(x\right) = {\lim}_{h \to 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\Rightarrow f ' \left(x\right) = {\lim}_{h \to 0} \frac{\frac{1}{3} \left(x + h\right) + \frac{4}{5} - \left(\frac{1}{3} x + \frac{4}{5}\right)}{h}$

$= {\lim}_{h \to 0} \frac{\cancel{\frac{1}{3} x} + \frac{1}{3} h \cancel{+ \frac{4}{5}} \cancel{- \frac{1}{3} x} \cancel{- \frac{4}{5}}}{h}$

$= {\lim}_{h \to 0} \frac{\frac{1}{3} {\cancel{h}}^{1}}{\cancel{h}} ^ 1 = \frac{1}{3}$