**Acceleration as a Function of Position**

To find the acceleration of the particle as a function of position, we need to differentiate the velocity function with respect to time.

Given that the velocity function is v(x) = β(x)^{-2n}, we can find the expression for acceleration, a(x), by taking the derivative of the velocity function with respect to time.

**Differentiating the Velocity Function**

To differentiate the velocity function, we need to use the chain rule of differentiation. Let's break down the steps:

1. Start with the velocity function: v(x) = β(x)^{-2n}.
2. Take the natural logarithm of both sides to simplify the differentiation process: ln(v(x)) = ln(β(x)^{-2n}).
3. Apply the power rule of logarithms to bring down the exponent: ln(v(x)) = -2n * ln(β(x)).
4. Differentiate both sides with respect to x using the chain rule:
- (1/v(x)) * dv(x)/dx = -2n * (1/β(x)) * dβ(x)/dx.
- Simplifying further, we get: dv(x)/dx = -2n * (v(x)/β(x)) * dβ(x)/dx.

**Expressing the Acceleration Function**

Now that we have the expression for dv(x)/dx, we can substitute the given velocity function v(x) = β(x)^{-2n} to find the acceleration function a(x).

1. Substitute the velocity function into the expression for dv(x)/dx:
dv(x)/dx = -2n * (β(x)^{-2n}/β(x)) * dβ(x)/dx.
2. Simplify the expression by combining the terms:
dv(x)/dx = -2n * (β(x)^{-2n-1}) * dβ(x)/dx.

Therefore, the acceleration function, a(x), is given by the expression:
a(x) = -2n * (β(x)^{-2n-1}) * dβ(x)/dx.

**Conclusion**

In summary, the acceleration of the particle as a function of position is given by the expression a(x) = -2n * (β(x)^{-2n-1}) * dβ(x)/dx. This is obtained by differentiating the velocity function v(x) = β(x)^{-2n} with respect to time using the chain rule. The resulting expression involves the terms β(x) and dβ(x)/dx, representing the constant β and its derivative with respect to position.

Beta*2nraised to power 2n-1