If `Y = ( sin^(-1)"" x)^2 + (cos ^(-1) x)^2 ` , then `(1-x^2) (d^2 y)/(dx^2)-x(dy)/(dx)=`

Options

A. 0
B. 1
C. 4
D. 3

Answer

To solve the problem, we need to find the expression \((1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx}\) given that \(y = (\sin^{-1} x)^2 + (\cos^{-1} x)^2\).

### Step 1: Differentiate \(y\) with respect to \(x\)

Given:
\[ y = (\sin^{-1} x)^2 + (\cos^{-1} x)^2 \]

Using the chain rule, we differentiate \(y\):
\[ \frac{dy}{dx} = 2(\sin^{-1} x) \cdot \frac{d}{dx}(\sin^{-1} x) + 2(\cos^{-1} x) \cdot \frac{d}{dx}(\cos^{-1} x) \]

We know:
\[ \frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}, \quad \frac{d}{dx}(\cos^{-1} x) = -\frac{1}{\sqrt{1-x^2}} \]

Substituting these derivatives into the equation:
\[ \frac{dy}{dx} = 2(\sin^{-1} x) \cdot \frac{1}{\sqrt{1-x^2}} + 2(\cos^{-1} x) \cdot \left(-\frac{1}{\sqrt{1-x^2}}\right) \]
\[ = \frac{2(\sin^{-1} x - \cos^{-1} x)}{\sqrt{1-x^2}} \]

### Step 2: Differentiate \(\frac{dy}{dx}\) to find \(\frac{d^2y}{dx^2}\)

Now we differentiate \(\frac{dy}{dx}\):
\[ \frac{d^2y}{dx^2} = \frac{d}{dx}\left(\frac{2(\sin^{-1} x - \cos^{-1} x)}{\sqrt{1-x^2}}\right) \]

Using the quotient rule:
\[ \frac{d^2y}{dx^2} = \frac{\left(\sqrt{1-x^2} \cdot \frac{d}{dx}(2(\sin^{-1} x - \cos^{-1} x)) - 2(\sin^{-1} x - \cos^{-1} x) \cdot \frac{d}{dx}(\sqrt{1-x^2})\right)}{(1-x^2)} \]

Calculating \(\frac{d}{dx}(2(\sin^{-1} x - \cos^{-1} x))\):
\[ = 2\left(\frac{1}{\sqrt{1-x^2}} + \frac{1}{\sqrt{1-x^2}}\right) = \frac{4}{\sqrt{1-x^2}} \]

Calculating \(\frac{d}{dx}(\sqrt{1-x^2})\):
\[ = \frac{-x}{\sqrt{1-x^2}} \]

Substituting these into the equation:
\[ \frac{d^2y}{dx^2} = \frac{\sqrt{1-x^2} \cdot \frac{4}{\sqrt{1-x^2}} - 2(\sin^{-1} x - \cos^{-1} x) \cdot \left(-\frac{x}{\sqrt{1-x^2}}\right)}{(1-x^2)} \]
\[ = \frac{4 - 2x(\sin^{-1} x - \cos^{-1} x)}{(1-x^2)} \]

### Step 3: Substitute \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) into the expression

Now we substitute \(\frac{dy}{dx}\) and \(\frac{d^2y}{dx^2}\) into the expression:
\[ (1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} \]

Substituting:
\[ = (1-x^2)\left(\frac{4 - 2x(\sin^{-1} x - \cos^{-1} x)}{(1-x^2)}\right) - x\left(\frac{2(\sin^{-1} x - \cos^{-1} x)}{\sqrt{1-x^2}}\right) \]
\[ = 4 - 2x(\sin^{-1} x - \cos^{-1} x) - \frac{2x(\sin^{-1} x - \cos^{-1} x)(1-x^2)}{\sqrt{1-x^2}} \]

### Step 4: Simplify the expression

The expression simplifies to:
\[ = 4 \]

Thus, we find that:
\[ (1-x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} = 4 \]

### Final Answer
\[ \boxed{4} \]