`y=sin^(-1)[sqrt(x-a x)-sqrt(a-a x)]`

Explanation

To find the derivative \( \frac{dy}{dx} \) for the function \( y = \sin^{-1} \left[ \sqrt{(x - ax)} - \sqrt{(a - ax)} \right] \), we can follow these steps:

### Step 1: Simplify the expression inside the inverse sine function

We start with:
\[ y = \sin^{-1} \left[ \sqrt{(x - ax)} - \sqrt{(a - ax)} \right] \]

Factor out \( x \) from the first square root:
\[ y = \sin^{-1} \left[ \sqrt{x(1 - a)} - \sqrt{a(1 - x)} \right] \]

### Step 2: Use the sine inverse subtraction formula

We can use the formula for the difference of two sine inverses:
\[ \sin^{-1}(a) - \sin^{-1}(b) = \sin^{-1}(a) - \sin^{-1}(b) \]
Let \( a = \sqrt{x(1 - a)} \) and \( b = \sqrt{a(1 - x)} \). Thus,
\[ y = \sin^{-1} \left( \sqrt{x(1 - a)} \right) - \sin^{-1} \left( \sqrt{a(1 - x)} \right) \]

### Step 3: Differentiate both sides with respect to \( x \)

Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx} \left[ \sin^{-1} \left( \sqrt{x(1 - a)} \right) \right] - \frac{d}{dx} \left[ \sin^{-1} \left( \sqrt{a(1 - x)} \right) \right] \]

Using the derivative of \( \sin^{-1}(u) \):
\[ \frac{d}{dx} \sin^{-1}(u) = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]

### Step 4: Calculate derivatives of the inner functions

For the first term:
Let \( u_1 = \sqrt{x(1 - a)} \):
\[ \frac{du_1}{dx} = \frac{1}{2\sqrt{x(1 - a)}} \cdot (1 - a) \]

For the second term:
Let \( u_2 = \sqrt{a(1 - x)} \):
\[ \frac{du_2}{dx} = \frac{1}{2\sqrt{a(1 - x)}} \cdot (-a) \]

### Step 5: Substitute back into the derivative

Now substituting back:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - x(1 - a)}} \cdot \frac{(1 - a)}{2\sqrt{x(1 - a)}} - \frac{1}{\sqrt{1 - a(1 - x)}} \cdot \frac{-a}{2\sqrt{a(1 - x)}} \]

### Step 6: Simplify the expression

This can be simplified further, but the key steps have been outlined. The final expression will depend on the specific values of \( a \) and \( x \).

### Final Answer

Thus, the derivative \( \frac{dy}{dx} \) can be expressed as:
\[ \frac{dy}{dx} = \frac{(1 - a)}{2\sqrt{x(1 - a)(1 - x(1 - a))}} + \frac{a}{2\sqrt{a(1 - x)(1 - a(1 - x))}} \]