Differentiate the following w.r.t. x :
`log((x+sqrt(x^(2)-a^(2)))/(x-sqrt(x^(2)-a^(2))))`

Answer

To differentiate the function \( y = \log\left(\frac{x + \sqrt{x^2 - a^2}}{x - \sqrt{x^2 - a^2}}\right) \) with respect to \( x \), we can follow these steps:

### Step 1: Rewrite the function
Let:
\[ y = \log\left(\frac{x + \sqrt{x^2 - a^2}}{x - \sqrt{x^2 - a^2}}\right) \]

### Step 2: Simplify using logarithmic properties
Using the property of logarithms that states \( \log\left(\frac{m}{n}\right) = \log(m) - \log(n) \), we can rewrite \( y \):
\[ y = \log(x + \sqrt{x^2 - a^2}) - \log(x - \sqrt{x^2 - a^2}) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}\left[\log(x + \sqrt{x^2 - a^2})\right] - \frac{d}{dx}\left[\log(x - \sqrt{x^2 - a^2})\right] \]

Using the chain rule, the derivative of \( \log(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{x + \sqrt{x^2 - a^2}} \cdot \left(1 + \frac{x}{\sqrt{x^2 - a^2}}\right) - \frac{1}{x - \sqrt{x^2 - a^2}} \cdot \left(1 - \frac{x}{\sqrt{x^2 - a^2}}\right) \]

### Step 4: Simplify the derivatives
Calculating the derivatives:
1. For \( \log(x + \sqrt{x^2 - a^2}) \):
\[ \frac{d}{dx}(x + \sqrt{x^2 - a^2}) = 1 + \frac{x}{\sqrt{x^2 - a^2}} \]

2. For \( \log(x - \sqrt{x^2 - a^2}) \):
\[ \frac{d}{dx}(x - \sqrt{x^2 - a^2}) = 1 - \frac{x}{\sqrt{x^2 - a^2}} \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{dy}{dx} = \frac{1 + \frac{x}{\sqrt{x^2 - a^2}}}{x + \sqrt{x^2 - a^2}} - \frac{1 - \frac{x}{\sqrt{x^2 - a^2}}}{x - \sqrt{x^2 - a^2}} \]

### Step 6: Finding a common denominator
To simplify this expression, we can find a common denominator:
\[ \frac{dy}{dx} = \frac{(1 + \frac{x}{\sqrt{x^2 - a^2}})(x - \sqrt{x^2 - a^2}) - (1 - \frac{x}{\sqrt{x^2 - a^2}})(x + \sqrt{x^2 - a^2})}{(x + \sqrt{x^2 - a^2})(x - \sqrt{x^2 - a^2})} \]

### Step 7: Simplifying the numerator
Expanding both terms in the numerator:
1. First term:
\[ (1 + \frac{x}{\sqrt{x^2 - a^2}})(x - \sqrt{x^2 - a^2}) = x - \sqrt{x^2 - a^2} + \frac{x^2}{\sqrt{x^2 - a^2}} - \frac{x \cdot (x^2 - a^2)}{\sqrt{x^2 - a^2}} \]

2. Second term:
\[ (1 - \frac{x}{\sqrt{x^2 - a^2}})(x + \sqrt{x^2 - a^2}) = x + \sqrt{x^2 - a^2} - \frac{x^2}{\sqrt{x^2 - a^2}} + \frac{x \cdot (x^2 - a^2)}{\sqrt{x^2 - a^2}} \]

### Step 8: Combine and simplify
After combining and simplifying, we will find that:
\[ \frac{dy}{dx} = \frac{2}{\sqrt{x^2 - a^2}} \]

### Final Result
Thus, the derivative of the given function is:
\[ \frac{dy}{dx} = \frac{2}{\sqrt{x^2 - a^2}} \]