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The problem asks for the derivative of \( f(x) = \frac{1}{x^{11}} \).

### Solution:
1. Rewrite the function using a negative exponent:
   \[
   f(x) = x^{-11}
   \]

2. Now, apply the **power rule** of differentiation. The power rule states that:
   \[
   \frac{d}{dx} \left( x^n \right) = n \cdot x^{n-1}
   \]
   For \( f(x) = x^{-11} \), we have:
   \[
   f'(x) = -11 \cdot x^{-12}
   \]

3. Therefore, the derivative is:
   \[
   f'(x) = \frac{-11}{x^{12}}
   \]

Would you like a detailed breakdown of any step, or have any questions on the topic?

### Related Questions:
1. How does the power rule apply to non-integer exponents?
2. What is the derivative of \( f(x) = x^n \) for different values of \( n \)?
3. How do negative exponents affect the behavior of a function and its derivative?
4. What other differentiation rules could be useful for more complex functions?
5. Can you provide the second derivative of \( f(x) = \frac{1}{x^{11}} \)?

**Tip:** Always simplify expressions by rewriting them in terms of powers before differentiating, especially when dealing with negative or fractional exponents.

Learn how to differentiate the function f(x) = 1/x^11 using the power rule. This calculus problem focuses on applying differentiation techniques to functions with negative exponents. Suitable for high school students studying calculus.

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