# How do you find the derivative of f(x)=x^2e^-x?

Feb 25, 2017

$f ' \left(x\right) = 2 x {e}^{-} x - {x}^{2} {e}^{-} x$

#### Explanation:

differentiate using the $\textcolor{b l u e}{\text{product rule}}$

$\text{Given "f(x)=g(x).h(x)" then}$

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

$\text{here } g \left(x\right) = {x}^{2} \Rightarrow g ' \left(x\right) = 2 x$

$\text{and } h \left(x\right) = {e}^{-} x \Rightarrow h ' \left(x\right) = {e}^{-} x . \frac{d}{\mathrm{dx}} \left(- x\right) = - {e}^{-} x$

$\Rightarrow f ' \left(x\right) = {x}^{2} \left(- {e}^{-} x\right) + {e}^{-} x \left(2 x\right)$

$\textcolor{w h i t e}{\Rightarrow f ' \left(x\right)} = 2 x {e}^{-} x - {x}^{2} {e}^{-} x$