if ` Y= log((1 + sqrt""x)/(1-sqrt""x ))` then ` (dy)/(dx) =`

Options

A. `(1)/(sqrt""x(1+x))`
B. `(1)/(sqrt"" x (1-x))`
C. `(sqrt""x)/(1+x)`
D. None

Answer

To find the derivative of \( Y = \log\left(\frac{1 + \sqrt{x}}{1 - \sqrt{x}}\right) \), we will use the chain rule and the quotient rule for differentiation. Let's go through the steps:

### Step 1: Differentiate using the chain rule
We start by applying the chain rule to differentiate \( Y \):
\[ \frac{dY}{dx} = \frac{1}{\frac{1 + \sqrt{x}}{1 - \sqrt{x}}} \cdot \frac{d}{dx}\left(\frac{1 + \sqrt{x}}{1 - \sqrt{x}}\right) \]

### Step 2: Differentiate the inner function using the quotient rule
Let \( u = 1 + \sqrt{x} \) and \( v = 1 - \sqrt{x} \). Then, we apply the quotient rule:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \]
Now we need to find \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
\[ \frac{du}{dx} = \frac{1}{2\sqrt{x}}, \quad \frac{dv}{dx} = -\frac{1}{2\sqrt{x}} \]

### Step 3: Substitute back into the quotient rule
Substituting \( u \), \( v \), \( \frac{du}{dx} \), and \( \frac{dv}{dx} \) into the quotient rule gives:
\[ \frac{d}{dx}\left(\frac{1 + \sqrt{x}}{1 - \sqrt{x}}\right) = \frac{(1 - \sqrt{x})\left(\frac{1}{2\sqrt{x}}\right) - (1 + \sqrt{x})\left(-\frac{1}{2\sqrt{x}}\right)}{(1 - \sqrt{x})^2} \]

### Step 4: Simplify the numerator
Now, simplify the numerator:
\[ = \frac{(1 - \sqrt{x})\left(\frac{1}{2\sqrt{x}}\right) + (1 + \sqrt{x})\left(\frac{1}{2\sqrt{x}}\right)}{(1 - \sqrt{x})^2} \]
\[ = \frac{\frac{1 - \sqrt{x} + 1 + \sqrt{x}}{2\sqrt{x}}}{(1 - \sqrt{x})^2} \]
\[ = \frac{\frac{2}{2\sqrt{x}}}{(1 - \sqrt{x})^2} = \frac{1}{\sqrt{x}(1 - \sqrt{x})^2} \]

### Step 5: Combine everything
Now substitute this back into the derivative of \( Y \):
\[ \frac{dY}{dx} = \frac{1}{\frac{1 + \sqrt{x}}{1 - \sqrt{x}}} \cdot \frac{1}{\sqrt{x}(1 - \sqrt{x})^2} \]
\[ = \frac{1 - \sqrt{x}}{1 + \sqrt{x}} \cdot \frac{1}{\sqrt{x}(1 - \sqrt{x})^2} \]

### Step 6: Final simplification
Now simplify this expression:
\[ = \frac{(1 - \sqrt{x})}{\sqrt{x}(1 + \sqrt{x})(1 - \sqrt{x})^2} \]
The \( (1 - \sqrt{x}) \) cancels out:
\[ = \frac{1}{\sqrt{x}(1 + \sqrt{x})(1 - \sqrt{x})} \]

Thus, the final answer is:
\[ \frac{dY}{dx} = \frac{1}{\sqrt{x}(1 - x)} \]