`(d)/(dx)[sin^(-1)(xsqrt(1 - x)- sqrt(x)sqrt(1 - x^(2)))]` is equal to

Options

A. `(1)/(2sqrt(x(1 -x)))-(1)/(sqrt(1-x^(2)))`
B. `(1)/(sqrt1-{xsqrt(1-x)-sqrt(x(1-x^(2))}^(2)}^(2))`
C. `(1)/(sqrt(1 -x^(2)))-(1)/(2sqrtx(1-x))`
D. `(1)/(sqrt(x(1-x)(1-x)^(2)))`

Explanation

To solve the problem, we need to differentiate the function given by:

\[ y = \sin^{-1}(x \sqrt{1 - x} - \sqrt{x} \sqrt{1 - x^2}) \]

### Step 1: Rewrite the Function
First, we can simplify the expression inside the inverse sine function. The expression can be rewritten as:

\[ y = \sin^{-1}(x \sqrt{1 - x} - \sqrt{x(1 - x^2)}) \]

### Step 2: Differentiate Using Chain Rule
To differentiate \(y\) with respect to \(x\), we will use the chain rule. The derivative of \(\sin^{-1}(u)\) is given by:

\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - u^2}} \cdot \frac{du}{dx} \]

where \(u = x \sqrt{1 - x} - \sqrt{x(1 - x^2)}\).

### Step 3: Differentiate \(u\)
Now, we need to find \(\frac{du}{dx}\). We differentiate \(u\):

\[ u = x \sqrt{1 - x} - \sqrt{x(1 - x^2)} \]

Using the product rule and chain rule, we differentiate each term in \(u\).

1. Differentiate \(x \sqrt{1 - x}\):
- Let \(v = \sqrt{1 - x}\), then \(\frac{dv}{dx} = -\frac{1}{2\sqrt{1 - x}}\).
- By product rule:
\[ \frac{d}{dx}(x \sqrt{1 - x}) = \sqrt{1 - x} + x \left(-\frac{1}{2\sqrt{1 - x}}\right) = \sqrt{1 - x} - \frac{x}{2\sqrt{1 - x}} \]

2. Differentiate \(-\sqrt{x(1 - x^2)}\):
- Let \(w = \sqrt{x(1 - x^2)}\).
- Using the product rule:
\[ \frac{d}{dx}(-\sqrt{x(1 - x^2)}) = -\frac{1}{2\sqrt{x(1 - x^2)}} \cdot \left( (1 - x^2) + x(-2x) \right) = -\frac{1 - 3x^2}{2\sqrt{x(1 - x^2)}} \]

### Step 4: Combine Derivatives
Now, we combine the derivatives:

\[ \frac{du}{dx} = \left(\sqrt{1 - x} - \frac{x}{2\sqrt{1 - x}}\right) - \frac{1 - 3x^2}{2\sqrt{x(1 - x^2)}} \]

### Step 5: Substitute Back into the Derivative of \(y\)
Now we substitute \(u\) and \(\frac{du}{dx}\) back into the derivative of \(y\):

\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 - (x \sqrt{1 - x} - \sqrt{x(1 - x^2)})^2}} \cdot \frac{du}{dx} \]

### Final Step: Simplify
Finally, we simplify the expression to get the final result.