The derivative of the function \( f(x) = \frac{1}{x} \) can be found using standard differentiation rules. This function can be rewritten as:

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

Now, apply the power rule for derivatives, which states:

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

For \( f(x) = x^{-1} \), the exponent \( n = -1 \), so:

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

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

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

Would you like a deeper explanation of this process or have any further questions?

Here are some related questions:
1. What is the derivative of \( \frac{1}{x^2} \)?
2. How do you find the derivative of a product involving \( \frac{1}{x} \)?
3. What is the second derivative of \( \frac{1}{x} \)?
4. How does the derivative of \( \frac{1}{x} \) behave as \( x \) approaches zero?
5. Can you explain the chain rule for composite functions involving \( \frac{1}{x} \)?

**Tip:** When applying the power rule, always reduce the exponent by 1 and multiply by the original exponent!

Find the derivative of 1/x using the power rule in calculus. This detailed guide walks through the steps of rewriting 1/x as x^-1 and applying the power rule to find f'(x) = -1/x^2. Ideal for students in Grades 11-12 or early college calculus courses.

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