# What is the derivative of  x^2 x e^-x?

May 25, 2018

$3 {x}^{2} {e}^{-} x - {x}^{3} {e}^{-} x$

#### Explanation:

Given: ${x}^{2} x {e}^{-} x$.

$= {x}^{3} {e}^{-} x$

We use the product rule, which states that,

$\frac{d}{\mathrm{dx}} \left(x y\right) = y \frac{d}{\mathrm{dx}} \left(x\right) + x \frac{d}{\mathrm{dx}} \left(y\right)$

$\therefore \frac{\mathrm{dy}}{\mathrm{dx}} \left({x}^{3} {e}^{-} x\right) = {e}^{-} x \frac{d}{\mathrm{dx}} \left({x}^{3}\right) + {x}^{3} \frac{d}{\mathrm{dx}} \left({e}^{-} x\right)$

$= {e}^{-} x \cdot 3 {x}^{2} + {x}^{3} \cdot - {e}^{-} x$

$= 3 {x}^{2} {e}^{-} x - {x}^{3} {e}^{-} x$