if `y=(1+1/x^(2))/(1-1/(x)^(2))`,then `(dy)/(dx)` is equal to

Options

A. `(-4x)/(x^(2)-1)^(2)`
B. `(-4x)/(x^(2)-1)`
C. `(1-x^(2))/(4x)`
D. `(4x)/(x^(2)-1)`

Explanation

To find the derivative of the function \( y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}} \), we will follow these steps:

### Step 1: Simplify the function
We start by simplifying the expression for \( y \).

\[ y = \frac{1 + \frac{1}{x^2}}{1 - \frac{1}{x^2}} \]

To simplify, we can take the least common multiple (LCM) of the numerator and denominator:

\[ y = \frac{\frac{x^2 + 1}{x^2}}{\frac{x^2 - 1}{x^2}} = \frac{x^2 + 1}{x^2 - 1} \]

### Step 2: Apply the Quotient Rule
Now, we need to differentiate \( y \) using the quotient rule. The quotient rule states that if \( y = \frac{u}{v} \), then:

\[ \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]

Here, let \( u = x^2 + 1 \) and \( v = x^2 - 1 \).

### Step 3: Differentiate \( u \) and \( v \)
Now we find the derivatives of \( u \) and \( v \):

\[ \frac{du}{dx} = 2x \]
\[ \frac{dv}{dx} = 2x \]

### Step 4: Substitute into the Quotient Rule
Now we substitute \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule formula:

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

### Step 5: Simplify the expression
Now we simplify the numerator:

\[ = \frac{2x(x^2 - 1) - 2x(x^2 + 1)}{(x^2 - 1)^2} \]

Distributing \( 2x \):

\[ = \frac{2x^3 - 2x - 2x^3 - 2x}{(x^2 - 1)^2} \]

Combining like terms:

\[ = \frac{-4x}{(x^2 - 1)^2} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:

\[ \frac{dy}{dx} = \frac{-4x}{(x^2 - 1)^2} \]