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We are tasked with finding the derivative of the function \( f(x) = \frac{7}{x^3} - \frac{3}{x^2} - \frac{1}{x} + 130 \).

### Step 1: Rewrite the function
To make differentiation easier, express the terms with negative exponents:

\[
f(x) = 7x^{-3} - 3x^{-2} - x^{-1} + 130
\]

### Step 2: Differentiate each term
We apply the power rule, which states:

\[
\frac{d}{dx} \left( x^n \right) = n x^{n-1}
\]

- For \( 7x^{-3} \), the derivative is \( 7 \cdot (-3) x^{-4} = -21x^{-4} \).
- For \( -3x^{-2} \), the derivative is \( -3 \cdot (-2) x^{-3} = 6x^{-3} \).
- For \( -x^{-1} \), the derivative is \( -1 \cdot (-1) x^{-2} = x^{-2} \).
- The derivative of the constant \( 130 \) is \( 0 \).

### Step 3: Combine the results
Putting all the terms together, we get:

\[
f'(x) = -21x^{-4} + 6x^{-3} + x^{-2}
\]

### Final Answer:
\[
f'(x) = -\frac{21}{x^4} + \frac{6}{x^3} + \frac{1}{x^2}
\]

Would you like more details or further explanation on any part of this process? 

Here are 5 related questions to consider:
1. How do you apply the power rule for more complex functions?
2. What is the derivative of a product of two functions?
3. How would you differentiate logarithmic functions?
4. What happens when differentiating functions involving square roots?
5. How does the chain rule apply in differentiation?

**Tip**: For any function with negative exponents, always rewrite them first to simplify differentiation using the power rule.

Find the derivative of the function f(x) = 7/x^3 - 3/x^2 - 1/x + 130 by using the rules of differentiation.

Learn how to differentiate the function f(x) = 7/x^3 - 3/x^2 - 1/x + 130 step by step using the power rule. This detailed solution is ideal for high school calculus students in grades 10-12.

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