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The problem asks you to use the definition of the derivative to show that if \( f(x) = \frac{1}{x} \), then \( f'(x) = -\frac{1}{x^2} \).

To solve this, let's proceed with the formal definition of a derivative, which is:

\[
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}
\]

Given \( f(x) = \frac{1}{x} \), we substitute this into the definition of the derivative:

\[
f'(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h}
\]

Now, let's simplify the expression inside the limit:

1. **Get a common denominator for the numerator:**

\[
\frac{1}{x+h} - \frac{1}{x} = \frac{x - (x + h)}{x(x + h)} = \frac{-h}{x(x+h)}
\]

2. **Substitute this back into the limit expression:**

\[
f'(x) = \lim_{h \to 0} \frac{\frac{-h}{x(x+h)}}{h}
\]

3. **Simplify the fraction:**

\[
f'(x) = \lim_{h \to 0} \frac{-h}{h \cdot x(x+h)} = \lim_{h \to 0} \frac{-1}{x(x+h)}
\]

4. **Take the limit as \( h \to 0 \):**

\[
f'(x) = \frac{-1}{x(x+0)} = \frac{-1}{x^2}
\]

Thus, the derivative of \( f(x) = \frac{1}{x} \) is indeed:

\[
f'(x) = -\frac{1}{x^2}
\]

Would you like further details or have any questions about this process?

Here are 5 follow-up questions you could explore:

1. How does the definition of the derivative relate to the concept of the slope of a tangent line?
2. What other methods, apart from the limit definition, can be used to compute derivatives?
3. How would the derivative change if \( f(x) = \frac{1}{x^n} \) for a general \( n \)?
4. Can you explain how the result \( f'(x) = -\frac{1}{x^2} \) behaves for different values of \( x \)?
5. How does this derivative relate to the power rule for negative exponents?

**Tip:** When dealing with fractions and derivatives, always remember that simplifying the algebra before taking the limit can save you time!

Use the definition of a derivative to show that if f(x) = 1/x, then f'(x) = -1/x^2.

Learn how to use the definition of a derivative to show that the derivative of f(x) = 1/x is f'(x) = -1/x^2. This step-by-step solution covers key calculus concepts, including the limit definition and algebraic simplification. Suitable for undergraduate students or AP Calculus learners.

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