A Nonlinear Spring Stores Potential Energy According To $U ( x ) = \alpha x ^ { 4 } + \beta x ^ { 2 }$

Multiple Choice

A nonlinear spring stores potential energy according to $U ( x ) = \alpha x ^ { 4 } + \beta x ^ { 2 }$ , where $\alpha$ and $\beta$ are constants and $x$ is the distance that the spring is either stretched or compressed. The force that would be required to compress this spring a distance $d$ is

A) $\alpha d ^ { 5 } / 5 + \beta d ^ { 3 } / 3$
B) $\alpha d ^ { 4 } + \beta d ^ { 2 }$
C) $2 d \left( 2 \alpha d ^ { 2 } + \beta \right)$
D) $4 \alpha d ^ { 4 } + 3 \beta d ^ { 3 }$