# Using the limit definition, how do you differentiate f(x)=x^2+3x+1?

Apr 11, 2018

$f ' \left(x\right) = 2 x + 3$

#### Explanation:

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

In this case

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

$= {\lim}_{h \rightarrow 0} \frac{{\left(x + h\right)}^{2} + 3 \left(x + h\right) + 1 - {x}^{2} - 3 x - 1}{h}$

$= {\lim}_{h \rightarrow 0} \frac{{x}^{2} + 2 x h + {h}^{2} + 3 x + 3 h + 1 - {x}^{2} - 3 x - 1}{h}$

$= {\lim}_{h \rightarrow 0} \frac{2 x h + {h}^{2} + 3 h}{h}$

$= {\lim}_{h \rightarrow 0} 2 x + h + 3 = 2 x + 3$.