The derivative of `f(x) = x^(-3) (3 + 7.x)` is

Options

A. `(9)/(x^(4) - (14)/(x^(3))`
B. `- (9)/(x^(4)) - 14 x^(3)`
C. `(9)/(x^(4)) + (14)/(x^(2))`
D. `-(9)/(x^(4)) - (14)/(x^(3))`

Answer

To find the derivative of the function \( f(x) = x^{-3}(3 + 7x) \), we will use the product rule and the power rule of differentiation.

### Step-by-step Solution:

1. **Identify the function**:
\[ f(x) = x^{-3}(3 + 7x) \]

2. **Apply the product rule**:
The product rule states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative \( f'(x) \) is given by:
\[ f'(x) = u'v + uv' \]
Here, let:
\[ u = x^{-3} \quad \text{and} \quad v = 3 + 7x \]

3. **Differentiate \( u \) and \( v \)**:
- For \( u = x^{-3} \):
\[ u' = -3x^{-4} \quad \text{(using the power rule)} \]
- For \( v = 3 + 7x \):
\[ v' = 7 \quad \text{(the derivative of a constant is 0)} \]

4. **Substitute into the product rule**:
Now substitute \( u \), \( u' \), \( v \), and \( v' \) into the product rule formula:
\[ f'(x) = (-3x^{-4})(3 + 7x) + (x^{-3})(7) \]

5. **Simplify the expression**:
- First term:
\[ -3x^{-4}(3 + 7x) = -9x^{-4} - 21x^{-3} \]
- Second term:
\[ (x^{-3})(7) = 7x^{-3} \]
- Combine the terms:
\[ f'(x) = -9x^{-4} - 21x^{-3} + 7x^{-3} \]
\[ f'(x) = -9x^{-4} - 14x^{-3} \]

6. **Final expression**:
Thus, the derivative of the function is:
\[ f'(x) = -\frac{9}{x^4} - \frac{14}{x^3} \]