Find the derivative of the function `(1 + (1)/(x)) (1 + (2)/( x))` with respect to x .

Answer

To find the derivative of the function \( f(x) = \left(1 + \frac{1}{x}\right) \left(1 + \frac{2}{x}\right) \), we can follow these steps:

### Step 1: Expand the Function
First, we will expand the function \( f(x) \).

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

Using the distributive property (also known as the FOIL method for binomials), we get:

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

This simplifies to:

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

Combining the terms, we have:

\[ f(x) = 1 + \frac{3}{x} + \frac{2}{x^2} \]

### Step 2: Rewrite in Terms of Powers of \( x \)
Next, we rewrite the function in terms of negative powers of \( x \):

\[ f(x) = 1 + 3x^{-1} + 2x^{-2} \]

### Step 3: Differentiate the Function
Now we differentiate \( f(x) \) with respect to \( x \):

\[ f'(x) = \frac{d}{dx}(1) + \frac{d}{dx}(3x^{-1}) + \frac{d}{dx}(2x^{-2}) \]

Using the power rule \( \frac{d}{dx}(x^n) = nx^{n-1} \):

\[ f'(x) = 0 + 3(-1)x^{-2} + 2(-2)x^{-3} \]

This simplifies to:

\[ f'(x) = -\frac{3}{x^2} - \frac{4}{x^3} \]

### Step 4: Combine the Derivative Terms
To combine the terms, we can express them with a common denominator:

\[ f'(x) = -\frac{3x}{x^3} - \frac{4}{x^3} = -\frac{3x + 4}{x^3} \]

### Final Answer
Thus, the derivative of the function is:

\[ f'(x) = -\frac{3x + 4}{x^3} \]