# How do you find the derivative of f(x)=x^3+x^2 using the limit process?

Nov 10, 2016

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

#### Explanation:

By definition of the derivative $f ' \left(x\right) = {\lim}_{h \rightarrow 0} \frac{f \left(x + h\right) - f \left(x\right)}{h}$
So with $f \left(x\right) = {x}^{3} + {x}^{2}$ we have;

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