Differentiate the following functions:
12. `8x^(3) - x^(2) + 5 - (2)/( x ) + (4)/( x^3)`.

Answer

To differentiate the function \( f(x) = 8x^3 - x^2 + 5 - \frac{2}{x} + \frac{4}{x^3} \), we will follow the rules of differentiation step by step.

### Step 1: Rewrite the function
First, we can rewrite the function in a more convenient form for differentiation by expressing the terms with negative exponents:
\[ f(x) = 8x^3 - x^2 + 5 - 2x^{-1} + 4x^{-3} \]

### Step 2: Differentiate each term
Now, we will differentiate each term separately using the power rule, which states that \( \frac{d}{dx}(x^n) = nx^{n-1} \).

1. Differentiate \( 8x^3 \):
\[ \frac{d}{dx}(8x^3) = 24x^2 \]

2. Differentiate \( -x^2 \):
\[ \frac{d}{dx}(-x^2) = -2x \]

3. Differentiate \( 5 \):
\[ \frac{d}{dx}(5) = 0 \]

4. Differentiate \( -2x^{-1} \):
\[ \frac{d}{dx}(-2x^{-1}) = 2x^{-2} \quad \text{(using the power rule)} \]

5. Differentiate \( 4x^{-3} \):
\[ \frac{d}{dx}(4x^{-3}) = -12x^{-4} \quad \text{(using the power rule)} \]

### Step 3: Combine the derivatives
Now we combine all the derivatives we calculated:
\[ f'(x) = 24x^2 - 2x + 0 + 2x^{-2} - 12x^{-4} \]

### Step 4: Rewrite in standard form
We can rewrite the derivative in a more standard form:
\[ f'(x) = 24x^2 - 2x + \frac{2}{x^2} - \frac{12}{x^4} \]

### Final Answer
Thus, the derivative of the function is:
\[ f'(x) = 24x^2 - 2x + \frac{2}{x^2} - \frac{12}{x^4} \]

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