`sin^-1[xsqrt(1-x)-sqrt(x)sqrt(1-x^2)]`

Answer

To solve the expression \( \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \), we will simplify it step by step.

### Step 1: Rewrite the Expression
Let \( y = \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \).

### Step 2: Simplify the Inside of the Inverse Sine
We can rewrite the expression inside the inverse sine:
\[ y = \sin^{-1}\left[x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2}\right] \]
Notice that \( \sqrt{1-x^2} = \sqrt{(1-x)(1+x)} \).

### Step 3: Factor the Expression
The expression can be factored:
\[ x\sqrt{1-x} - \sqrt{x}\sqrt{1-x^2} = \sqrt{x}\left(\sqrt{1-x} - \sqrt{1-x^2}\right) \]

### Step 4: Use the Identity for \( \sin^{-1} \)
We can use the identity:
\[ \sin^{-1}(a) - \sin^{-1}(b) = \sin^{-1}\left(a\sqrt{1-b^2} - b\sqrt{1-a^2}\right) \]
In our case, we can let \( a = x \) and \( b = \sqrt{x} \).

### Step 5: Apply the Identity
Using the identity, we can write:
\[ y = \sin^{-1}(x) - \sin^{-1}(\sqrt{x}) \]

### Step 6: Differentiate the Expression
Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(\sin^{-1}(x)) - \frac{d}{dx}(\sin^{-1}(\sqrt{x})) \]
Using the derivative of \( \sin^{-1}(u) \):
\[ \frac{d}{dx}(\sin^{-1}(u)) = \frac{1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \]
For \( \sin^{-1}(x) \):
\[ \frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}} \]
For \( \sin^{-1}(\sqrt{x}) \):
\[ \frac{d}{dx}(\sin^{-1}(\sqrt{x})) = \frac{1}{\sqrt{1-x}} \cdot \frac{1}{2\sqrt{x}} = \frac{1}{2\sqrt{x}\sqrt{1-x}} \]

### Step 7: Combine the Derivatives
Combining the derivatives gives us:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}} - \frac{1}{2\sqrt{x}\sqrt{1-x}} \]

### Final Result
Thus, the derivative of the given expression is:
\[ \frac{dy}{dx} = \frac{1}{\sqrt{1-x^2}} - \frac{1}{2\sqrt{x}\sqrt{1-x}} \]