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

Options

A. ` (2)/(sqrt(x^(2)+a^(2)))`
B. ` (-2)/(sqrt(x^(2)+a^(2)))`
C. ` (2x)/(sqrt(x^(2)+a^(2)))`
D. ` (-2x)/(sqrt(x^(2)+a^(2)))`

Answer

To solve the problem \( y = \log\left(\frac{x + \sqrt{x^2 + a^2}}{-x + \sqrt{x^2 + a^2}}\right) \), we will follow these steps:

### Step 1: Rewrite the Logarithm
We can rewrite the expression inside the logarithm:
\[ y = \log\left(\frac{x + \sqrt{x^2 + a^2}}{-x + \sqrt{x^2 + a^2}}\right) \]
This can be expressed as:
\[ y = \log\left(\frac{\sqrt{x^2 + a^2} + x}{\sqrt{x^2 + a^2} - x}\right) \]

### Step 2: Rationalize the Denominator
To simplify, we multiply the numerator and denominator by the conjugate of the denominator:
\[ y = \log\left(\frac{(\sqrt{x^2 + a^2} + x)(\sqrt{x^2 + a^2} + x)}{(\sqrt{x^2 + a^2} - x)(\sqrt{x^2 + a^2} + x)}\right) \]
The denominator simplifies to:
\[ (\sqrt{x^2 + a^2})^2 - x^2 = a^2 \]
So, we have:
\[ y = \log\left(\frac{(\sqrt{x^2 + a^2} + x)^2}{a^2}\right) \]

### Step 3: Apply Logarithmic Properties
Using the properties of logarithms:
\[ y = \log\left((\sqrt{x^2 + a^2} + x)^2\right) - \log(a^2) \]
This simplifies to:
\[ y = 2\log(\sqrt{x^2 + a^2} + x) - 2\log(a) \]

### Step 4: Differentiate
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 2 \cdot \frac{1}{\sqrt{x^2 + a^2} + x} \cdot \frac{d}{dx}(\sqrt{x^2 + a^2} + x) \]
The derivative of \( \sqrt{x^2 + a^2} \) is:
\[ \frac{d}{dx}(\sqrt{x^2 + a^2}) = \frac{x}{\sqrt{x^2 + a^2}} \]
Thus:
\[ \frac{d}{dx}(\sqrt{x^2 + a^2} + x) = \frac{x}{\sqrt{x^2 + a^2}} + 1 \]
Putting it all together:
\[ \frac{dy}{dx} = 2 \cdot \frac{1}{\sqrt{x^2 + a^2} + x} \cdot \left(\frac{x}{\sqrt{x^2 + a^2}} + 1\right) \]

### Step 5: Simplify
This can be simplified further:
\[ \frac{dy}{dx} = \frac{2\left(\frac{x + \sqrt{x^2 + a^2}}{\sqrt{x^2 + a^2}}\right)}{\sqrt{x^2 + a^2} + x} \]
The \( \sqrt{x^2 + a^2} + x \) cancels out:
\[ \frac{dy}{dx} = \frac{2}{\sqrt{x^2 + a^2}} \]

### Final Answer
Thus, the derivative is:
\[ \frac{dy}{dx} = \frac{2}{\sqrt{x^2 + a^2}} \]