If `y = log ((sqrt(1 + x^(2)) + x)/(sqrt(1 + x^(2)) -x)) " then " (dy)/(dx) =` ?

Options

A. `(2)/(sqrt(1 + x^(2)))`
B. `(2 sqrt(1 + x^(2)))/(x^(2))`
C. `(-2)/(sqrt(1 +x^(2)))`
D. none of these

Explanation

To find the derivative of the given function \( y = \log \left( \frac{\sqrt{1 + x^2} + x}{\sqrt{1 + x^2} - x} \right) \), we will follow these steps:

### Step 1: Use the property of logarithms
We can simplify the logarithm using the property \( \log \left( \frac{A}{B} \right) = \log A - \log B \).

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

### Step 2: Differentiate both sides
Now we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \frac{d}{dx} \left( \log \left( \sqrt{1 + x^2} + x \right) \right) - \frac{d}{dx} \left( \log \left( \sqrt{1 + x^2} - x \right) \right) \]

Using the derivative of \( \log u \) which is \( \frac{1}{u} \frac{du}{dx} \):

\[ \frac{dy}{dx} = \frac{1}{\sqrt{1 + x^2} + x} \cdot \frac{d}{dx} \left( \sqrt{1 + x^2} + x \right) - \frac{1}{\sqrt{1 + x^2} - x} \cdot \frac{d}{dx} \left( \sqrt{1 + x^2} - x \right) \]

### Step 3: Differentiate the inner functions
Now we need to differentiate \( \sqrt{1 + x^2} + x \) and \( \sqrt{1 + x^2} - x \):

1. For \( \sqrt{1 + x^2} + x \):
\[ \frac{d}{dx} \left( \sqrt{1 + x^2} + x \right) = \frac{1}{2\sqrt{1 + x^2}} \cdot 2x + 1 = \frac{x}{\sqrt{1 + x^2}} + 1 \]

2. For \( \sqrt{1 + x^2} - x \):
\[ \frac{d}{dx} \left( \sqrt{1 + x^2} - x \right) = \frac{1}{2\sqrt{1 + x^2}} \cdot 2x - 1 = \frac{x}{\sqrt{1 + x^2}} - 1 \]

### Step 4: Substitute back into the derivative
Now substituting these derivatives back into our expression for \( \frac{dy}{dx} \):

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

### Step 5: Simplify the expression
Now we simplify the expression:

\[ \frac{dy}{dx} = \frac{\frac{x}{\sqrt{1 + x^2}} + 1}{\sqrt{1 + x^2} + x} - \frac{\frac{x}{\sqrt{1 + x^2}} - 1}{\sqrt{1 + x^2} - x} \]

### Step 6: Combine the fractions
To combine the two fractions, we will find a common denominator:

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

### Step 7: Final simplification
After simplifying the numerator and denominator, we will arrive at the final expression for \( \frac{dy}{dx} \).

Finally, the result is:

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