Find the derivatives of the following at any point of their domains :
`g(x)=(1+1/x)/(1-1/x)`

Answer

To find the derivative of the function \( g(x) = \frac{1 + \frac{1}{x}}{1 - \frac{1}{x}} \), we will follow these steps:

### Step 1: Simplify the function
First, let's rewrite \( g(x) \) in a simpler form. We can multiply the numerator and the denominator by \( x \) to eliminate the fractions:

\[ g(x) = \frac{(1 + \frac{1}{x}) \cdot x}{(1 - \frac{1}{x}) \cdot x} = \frac{x + 1}{x - 1} \]

### Step 2: Apply the quotient rule
Now we will find the derivative \( g'(x) \) using the quotient rule. The quotient rule states that if \( g(x) = \frac{u}{v} \), then:

\[ g'(x) = \frac{u'v - uv'}{v^2} \]

Here, let \( u = x + 1 \) and \( v = x - 1 \).

### Step 3: Find \( u' \) and \( v' \)
We need to find the derivatives of \( u \) and \( v \):

\[ u' = \frac{d}{dx}(x + 1) = 1 \]

\[ v' = \frac{d}{dx}(x - 1) = 1 \]

### Step 4: Substitute into the quotient rule
Now we substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula:

\[ g'(x) = \frac{(1)(x - 1) - (x + 1)(1)}{(x - 1)^2} \]

### Step 5: Simplify the numerator
Now, simplify the numerator:

\[ g'(x) = \frac{x - 1 - (x + 1)}{(x - 1)^2} = \frac{x - 1 - x - 1}{(x - 1)^2} = \frac{-2}{(x - 1)^2} \]

### Final Result
Thus, the derivative of the function \( g(x) \) is:

\[ g'(x) = \frac{-2}{(x - 1)^2} \]

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