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

Options

A. 1
B. `-1`
C. `-1/2`
D. `1/2`

Answer

To solve the problem, we need to find the expression \( \frac{y^2}{2} + \frac{dy}{dx} \) where \( y = \frac{e^x + 1}{e^x - 1} \).

### Step 1: Find \( y^2 \)
First, we will calculate \( y^2 \):

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

Squaring \( y \):

\[ y^2 = \left(\frac{e^x + 1}{e^x - 1}\right)^2 = \frac{(e^x + 1)^2}{(e^x - 1)^2} \]

### Step 2: Find \( \frac{dy}{dx} \)
Now, we differentiate \( y \) with respect to \( x \) using the quotient rule:

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

Calculating the derivatives:

\[ \frac{d}{dx}(e^x + 1) = e^x \]
\[ \frac{d}{dx}(e^x - 1) = e^x \]

Substituting back into the quotient rule:

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

Simplifying the numerator:

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

### Step 3: Calculate \( \frac{y^2}{2} \)
Now, we calculate \( \frac{y^2}{2} \):

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

### Step 4: Combine \( \frac{y^2}{2} \) and \( \frac{dy}{dx} \)
Now we add \( \frac{y^2}{2} \) and \( \frac{dy}{dx} \):

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

Finding a common denominator:

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

### Step 5: Simplify the numerator
We expand the numerator:

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

Now substituting back:

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

This can be factored as:

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

### Step 6: Final simplification
The \( (e^x - 1)^2 \) terms cancel out:

\[ = \frac{1}{2} \]

### Final Answer
Thus, we have:

\[ \frac{y^2}{2} + \frac{dy}{dx} = \frac{1}{2} \]