Differentiate w.r.t `'x' f(x) = (sqrt(x^(2)+1)+sqrt(x^(2)-1))/(sqrt(x^(2)+1)-sqrt(x^(2)-1))`

Answer

To differentiate the function \( f(x) = \frac{\sqrt{x^2 + 1} + \sqrt{x^2 - 1}}{\sqrt{x^2 + 1} - \sqrt{x^2 - 1}} \) with respect to \( x \), we will follow these steps:

### Step 1: Rationalize the Denominator
To simplify the function, we can rationalize the denominator. We multiply the numerator and denominator by the conjugate of the denominator:

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

This gives us:

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

### Step 2: Simplify the Expression
Now, simplify the numerator and denominator:

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

This simplifies to:

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

### Step 3: Further Simplification
The denominator can be simplified using the difference of squares:

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

Thus, we have:

\[ f(x) = \frac{2}{2} = 1 \]

### Step 4: Differentiate the Function
Now that we have simplified \( f(x) \) to a constant \( 1 \), we can differentiate it:

\[ f'(x) = 0 \]

### Final Answer
The derivative of the function \( f(x) \) with respect to \( x \) is:

\[ f'(x) = 0 \]