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.