The derivative of `f(x) = (x + (1)/(x))^(3)`

Options

A. `3 x^(2) + (3)/(x^(4)) - 3`
B. `3 x^(2) - (3)/(x^(4)) + 3 - (3)/(x^(2))`
C. `3x^(2) + (3)/(x^(4)) - 3 + (3)/(x^(2))`
D. `3x^(2) + (3)/(x^(4)) 3 + (3)/(x^(2))`

Answer

To find the derivative of the function \( f(x) = \left( x + \frac{1}{x} \right)^3 \), we can follow these steps:

### Step 1: Expand the Function
We start by expanding the expression using the binomial theorem for \( (a + b)^3 \):
\[ f(x) = (x + \frac{1}{x})^3 = x^3 + 3x^2 \cdot \frac{1}{x} + 3x \cdot \left(\frac{1}{x}\right)^2 + \left(\frac{1}{x}\right)^3 \]
This simplifies to:
\[ f(x) = x^3 + 3x + \frac{3}{x} + \frac{1}{x^3} \]

### Step 2: Rewrite the Function
We can rewrite the function in a more convenient form for differentiation:
\[ f(x) = x^3 + 3x + 3x^{-1} + x^{-3} \]

### Step 3: Differentiate the Function
Now we differentiate \( f(x) \) term by term:
1. The derivative of \( x^3 \) is \( 3x^2 \).
2. The derivative of \( 3x \) is \( 3 \).
3. The derivative of \( 3x^{-1} \) is \( -3x^{-2} \) (using the power rule).
4. The derivative of \( x^{-3} \) is \( -3x^{-4} \).

Putting it all together, we have:
\[ f'(x) = 3x^2 + 3 - 3x^{-2} - 3x^{-4} \]

### Step 4: Simplify the Derivative
We can rewrite the derivative in a more standard form:
\[ f'(x) = 3x^2 + 3 - \frac{3}{x^2} - \frac{3}{x^4} \]

### Final Answer
Thus, the derivative of \( f(x) = \left( x + \frac{1}{x} \right)^3 \) is:
\[ f'(x) = 3x^2 + 3 - \frac{3}{x^2} - \frac{3}{x^4} \]

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