The derivative of \( \frac{1}{x} \) with respect to \( x \) is found using the power rule. We can rewrite \( \frac{1}{x} \) as \( x^{-1} \). Then, using the power rule \( \frac{d}{dx}[x^n] = nx^{n-1} \), we proceed as follows:

\[
\frac{d}{dx}\left( x^{-1} \right) = -1 \cdot x^{-2} = -\frac{1}{x^2}.
\]

So, the derivative of \( \frac{1}{x} \) with respect to \( x \) is:

\[
\frac{d}{dx} \left( \frac{1}{x} \right) = -\frac{1}{x^2}.
\]

Would you like more details or have any questions?

Here are 5 related questions to explore this topic further:
1. What is the derivative of \( \frac{1}{x^2} \)?
2. How would you find the second derivative of \( \frac{1}{x} \)?
3. What is the integral of \( \frac{1}{x} \) with respect to \( x \)?
4. How does the derivative of \( \frac{1}{x} \) change if \( x \) is a function of another variable?
5. What is the derivative of \( \frac{1}{x} \) with respect to a different variable, say \( t \), where \( x = f(t) \)?

**Tip:** When differentiating functions like \( \frac{1}{x} \), rewriting them using exponents (e.g., \( x^{-1} \)) often simplifies the differentiation process.

Explore how to find the derivative of 1/x with respect to x using the power rule. This solution demonstrates key calculus concepts and formulas for differentiating functions like 1/x. Suitable for students in Grades 10-12.

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