Given equation:
Dx/√(2ax - x^2) = a^n sin^(-1)(x/a - 1)

To find:
The value of n

Approach:
To determine the value of n, we need to simplify the given equation and compare it with a standard form of the derivative of inverse trigonometric function.

Solution:

Step 1: Simplify the given equation

Let's start by simplifying the left-hand side (LHS) of the equation:

Dx/√(2ax - x^2)

We can rewrite the denominator as (x-a)^2:

Dx/√(a^2 - (x-a)^2)

Using the formula Dx/√(a^2 - x^2) = 1/√(a^2 - x^2), we can rewrite the LHS as:

1/√(a^2 - (x-a)^2)

Step 2: Compare with the standard form

Now, let's compare the simplified LHS with the standard form of the derivative of inverse trigonometric function:

1/√(a^2 - (x-a)^2) = 1/√(a^2 - x^2)

We can observe that the denominator of the simplified LHS matches the denominator of the standard form. However, the numerator is different.

Step 3: Finding the value of n

To make the numerator of the simplified LHS match the numerator of the standard form, we need to rewrite it as:

1/√(a^2 - x^2) * (x-a)

Now, let's compare the simplified LHS with the standard form again:

1/√(a^2 - x^2) * (x-a) = a^n

From the above comparison, we can conclude that:

n = 1

Therefore, the value of n is 1.

Summary:
The value of n in the given equation Dx/√(2ax - x^2) = a^n sin^(-1)(x/a - 1) is 1. By simplifying the equation and comparing it with the standard form of the derivative of inverse trigonometric function, we can determine that n equals 1.