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

Options

A. `(e^(x))/((1-e^(x))*sqrt(1-e^(2x)))`
B. `(e^(x))/((1-e^(x))*sqrt(1-e^(x)))`
C. `(e^(x))/((1-e^(x))*(1+e^(2x)))`
D. `(e^(x))/((1-e^(x))*sqrt(1+e^(x)))`

Answer

To find the derivative of the function \( y = \sqrt{\frac{1 + e^x}{1 - e^x}} \), we will use the chain rule and the quotient rule. Here’s the step-by-step solution:

### Step 1: Rewrite the function
We can express \( y \) in a more manageable form:
\[ y = \left( \frac{1 + e^x}{1 - e^x} \right)^{1/2} \]

### Step 2: Apply the chain rule
Using the chain rule, we differentiate \( y \):
\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{1 + e^x}{1 - e^x} \right)^{-1/2} \cdot \frac{d}{dx} \left( \frac{1 + e^x}{1 - e^x} \right) \]

### Step 3: Differentiate the quotient
Now we need to differentiate \( \frac{1 + e^x}{1 - e^x} \) using the quotient rule. The quotient rule states that if \( u = f(x) \) and \( v = g(x) \), then:
\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Here, \( u = 1 + e^x \) and \( v = 1 - e^x \).

Calculating \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
\[ \frac{du}{dx} = e^x, \quad \frac{dv}{dx} = -e^x \]

Now applying the quotient rule:
\[ \frac{d}{dx} \left( \frac{1 + e^x}{1 - e^x} \right) = \frac{(1 - e^x)e^x - (1 + e^x)(-e^x)}{(1 - e^x)^2} \]

### Step 4: Simplify the numerator
Expanding the numerator:
\[ (1 - e^x)e^x + (1 + e^x)e^x = e^x - e^{2x} + e^x + e^{2x} = 2e^x \]

So we have:
\[ \frac{d}{dx} \left( \frac{1 + e^x}{1 - e^x} \right) = \frac{2e^x}{(1 - e^x)^2} \]

### Step 5: Substitute back into the derivative
Now substituting this back into our expression for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2} \left( \frac{1 + e^x}{1 - e^x} \right)^{-1/2} \cdot \frac{2e^x}{(1 - e^x)^2} \]

### Step 6: Simplify the expression
The expression simplifies to:
\[ \frac{dy}{dx} = \frac{e^x}{(1 - e^x)^2} \cdot \left( \frac{1 - e^x}{1 + e^x} \right)^{1/2} \]

### Final Result
Thus, the derivative is:
\[ \frac{dy}{dx} = \frac{e^x}{(1 - e^x)^{3/2} \cdot (1 + e^x)^{1/2}} \]

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