If `y=e^(2 sin ^(-1)x)` then `|((x ^(2) -1) y ^('') +xy')/(y)|` is equal to

Answer

To solve the problem step by step, we start with the function given:

**Step 1: Define the function**
Given \( y = e^{2 \sin^{-1}(x)} \).

**Step 2: Differentiate \( y \) to find \( y' \)**
Using the chain rule, we differentiate \( y \):
\[ y' = \frac{d}{dx}(e^{2 \sin^{-1}(x)}) = e^{2 \sin^{-1}(x)} \cdot \frac{d}{dx}(2 \sin^{-1}(x)) = e^{2 \sin^{-1}(x)} \cdot \frac{2}{\sqrt{1 - x^2}}. \]
Thus,
\[ y' = \frac{2 e^{2 \sin^{-1}(x)}}{\sqrt{1 - x^2}}. \]

**Step 3: Differentiate \( y' \) to find \( y'' \)**
Now we differentiate \( y' \):
\[ y'' = \frac{d}{dx}\left(\frac{2 e^{2 \sin^{-1}(x)}}{\sqrt{1 - x^2}}\right). \]
Using the quotient rule:
\[ y'' = \frac{(u'v - uv')}{v^2}, \]
where \( u = 2 e^{2 \sin^{-1}(x)} \) and \( v = \sqrt{1 - x^2} \).

Calculating \( u' \):
\[ u' = 2 \cdot e^{2 \sin^{-1}(x)} \cdot \frac{2}{\sqrt{1 - x^2}} = \frac{4 e^{2 \sin^{-1}(x)}}{\sqrt{1 - x^2}}. \]

Calculating \( v' \):
\[ v' = \frac{-x}{\sqrt{1 - x^2}}. \]

Now substituting into the quotient rule:
\[ y'' = \frac{\left(\frac{4 e^{2 \sin^{-1}(x)}}{\sqrt{1 - x^2}} \cdot \sqrt{1 - x^2} - 2 e^{2 \sin^{-1}(x)} \cdot \left(-\frac{x}{\sqrt{1 - x^2}}\right)\right)}{1 - x^2}. \]
This simplifies to:
\[ y'' = \frac{4 e^{2 \sin^{-1}(x)} + 2 x e^{2 \sin^{-1}(x)}}{(1 - x^2)^{3/2}}. \]
Thus,
\[ y'' = \frac{2 e^{2 \sin^{-1}(x)} (2 + x)}{(1 - x^2)^{3/2}}. \]

**Step 4: Substitute into the expression**
Now we substitute \( y \), \( y' \), and \( y'' \) into the expression:
\[ \left| \frac{(x^2 - 1) y'' + x y'}{y} \right|. \]
Substituting:
\[ = \left| \frac{(x^2 - 1) \cdot \frac{2 e^{2 \sin^{-1}(x)} (2 + x)}{(1 - x^2)^{3/2}} + x \cdot \frac{2 e^{2 \sin^{-1}(x)}}{\sqrt{1 - x^2}}}{e^{2 \sin^{-1}(x)}} \right|. \]
This simplifies to:
\[ = \left| \frac{2 e^{2 \sin^{-1}(x)} \left( (x^2 - 1)(2 + x) \cdot \frac{1}{(1 - x^2)^{3/2}} + x \cdot \frac{1}{\sqrt{1 - x^2}} \right)}{e^{2 \sin^{-1}(x)}} \right|. \]
The \( e^{2 \sin^{-1}(x)} \) cancels out:
\[ = \left| 2 \left( \frac{(x^2 - 1)(2 + x)}{(1 - x^2)^{3/2}} + \frac{x}{\sqrt{1 - x^2}} \right) \right|. \]

**Step 5: Simplify the expression**
This can be simplified further:
\[ = 2 \left| \frac{(x^2 - 1)(2 + x) + x(1 - x^2)}{(1 - x^2)^{3/2}} \right|. \]
This leads to:
\[ = 2 \left| \frac{2x^2 - 2 + x - x^3}{(1 - x^2)^{3/2}} \right|. \]

Thus, the final answer is:
\[ = \frac{2(2 - x^2 + x)}{(1 - x^2)^{3/2}}. \]