What are the critical points of f(x) =e^x-x^2e^(x^2)?

Jul 24, 2017

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

Explanation:

$\left({e}^{x}\right) ' = {e}^{x}$
$\left({e}^{u}\right) ' = {e}^{u} \cdot u '$

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

use the Product Rule $f ' g + f g '$ for ${x}^{2} \cdot {e}^{{x}^{2}}$
$f ' \left(x\right) = {e}^{x} - \left(2 x\right) \left({e}^{{x}^{2}}\right) + \left({e}^{x}\right) \left(2 x {e}^{{x}^{2}}\right)$
$f ' \left(x\right) = {e}^{x} - 2 x {e}^{{x}^{2}} + {e}^{x} \left(2 x {e}^{{x}^{2}}\right)$