Acceleration ($a$) is defined as the rate of change of velocity with respect to time ($a = \frac{dv}{dt}$). When velocity is given as a function of position $v(x)$, we use the chain rule to express acceleration as: $a = \frac{dv}{dx} \cdot \frac{dx}{dt} = v \frac{dv}{dx}$

1. Differentiate velocity with respect to position: Given $v = \beta x^{-2n}$ $\frac{dv}{dx} = \frac{d}{dx}(\beta x^{-2n}) = \beta(-2n)x^{-2n-1} = -2n\beta x^{-(2n+1)}$

2. Substitute into the acceleration formula: $a = v \left( \frac{dv}{dx} \right)$ $a = (\beta x^{-2n}) \cdot (-2n\beta x^{-(2n+1)})$

3. Simplify: $a = -2n\beta^2 x^{-2n + (-2n - 1)}$ $a = -2n\beta^2 x^{-4n - 1}$