# How do you use the definition of a derivative to find the derivative of f(x)=7x+4?

Dec 6, 2016

$f ' \left(x\right) = 7$

#### Explanation:

The derivative from $\textcolor{b l u e}{\text{first principles}}$ is found using.

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

The aim being to eliminate h from the denominator so there is no undefined situation.

$\Rightarrow f ' \left(x\right) = {\lim}_{h \to 0} \frac{7 \left(x + h\right) + 4 - \left(7 x + 4\right)}{h}$

$= {\lim}_{h \to 0} \frac{7 x + 7 h + 4 - 7 x - 4}{h}$

$= {\lim}_{h \to 0} \frac{7 \cancel{h}}{\cancel{h}} = 7$

$\Rightarrow f ' \left(x\right) = 7$