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

Options

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

Answer

To find the derivative \( \frac{dy}{dx} \) for the function given by

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

we can follow these steps:

### Step 1: Simplify the function
First, we simplify the expression for \( y \):

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

To simplify, we can combine the terms in the numerator and the denominator:

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

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

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

where \( u = x^2 + 1 \) and \( v = x^2 - 1 \).

### Step 3: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)
Now we calculate the derivatives of \( u \) and \( v \):

\[ \frac{du}{dx} = \frac{d}{dx}(x^2 + 1) = 2x, \]

\[ \frac{dv}{dx} = \frac{d}{dx}(x^2 - 1) = 2x. \]

### Step 4: Apply the quotient rule
Now substituting \( u, v, \frac{du}{dx}, \) 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:

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

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

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

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

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