Answer
$e^{^{\frac{-x}{4}}}(4 -x) $
Work Step by Step
We can find the negative gradient of the potential energy by solving for the force applied:
$F(x) =- \frac{dU(x)}{dx}$ ( The force applied is equal to the negative derivitive of the potential energy)
$=- \frac{d}{dx}(-4xe^{^{\frac{-x}{4}}})$
$=4(( \frac{-x}{4}\times e^{^{\frac{-x}{4}}}) + e^{^{\frac{-x}{4}}})$
$=( {-x} e^{^{\frac{-x}{4}}}) + 4e^{^{\frac{-x}{4}}}$
$=4e^{^{\frac{-x}{4}}} -x e^{^{\frac{-x}{4}}} $
$=e^{^{\frac{-x}{4}}}(4 -x) $
Therefore the formula for the negative gradient of the potential energy is $e^{^{\frac{-x}{4}}}(4 -x) $
