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

Options

A. `1+y^(2)`
B. `1-y^(2)`
C. `y^(2)-1`
D. `-(1+y^(2))`

Answer

To find the derivative \( \frac{dy}{dx} \) of the function \( y = \frac{e^{2x} - 1}{e^{2x} + 1} \), we will use the quotient rule of differentiation. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), then the derivative is given by:

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

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

### Step 1: Identify \( u \) and \( v \)

Let:
- \( u = e^{2x} - 1 \)
- \( v = e^{2x} + 1 \)

### Step 2: Find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \)

Now, we need to compute the derivatives of \( u \) and \( v \):

1. **Derivative of \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(e^{2x} - 1) = 2e^{2x} \]

2. **Derivative of \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(e^{2x} + 1) = 2e^{2x} \]

### Step 3: Apply the Quotient Rule

Now, we can apply the quotient rule:

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

### Step 4: Simplify the Expression

Now we simplify the numerator:

1. Expand the numerator:
\[ = (2e^{2x}(e^{2x} + 1)) - (2e^{2x}(e^{2x} - 1)) \]
\[ = 2e^{4x} + 2e^{2x} - (2e^{4x} - 2e^{2x}) \]
\[ = 2e^{4x} + 2e^{2x} - 2e^{4x} + 2e^{2x} \]
\[ = 4e^{2x} \]

2. Thus, the derivative becomes:
\[ \frac{dy}{dx} = \frac{4e^{2x}}{(e^{2x} + 1)^2} \]

### Final Answer

The derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{4e^{2x}}{(e^{2x} + 1)^2} \]

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