Question Statement

Find $\frac{dy}{dx}$ given that $y=\frac{1}{x}$.

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Background and Explanation

This problem requires differentiating a reciprocal function using the power rule. You will first need to rewrite the fraction $\frac{1}{x}$ as a negative exponent $x^{-1}$ to apply the standard differentiation formula.

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Solution

We are given the function: $y=\frac{1}{x}$

To find the derivative, we differentiate both sides with respect to $x$:

$\begin{array}{rl}\frac{dy}{dx}& =\frac{d}{dx}\left(\frac{1}{x}\right)\\ & =\frac{d}{dx}\left(x^{-1}\right)\\ & =(-1)x^{-1-1}\\ \frac{dy}{dx}& =(-1)x^{-2}=\frac{-1}{x^2}\end{array}$

Therefore, the derivative is: $\frac{dy}{dx}=-\frac{1}{x^2}$

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Key Formulas or Methods Used

• Power Rule for Differentiation: $\frac{d}{dx}(x^n)=n{x}^{n-1}$
• Negative Exponent Rule: $\frac{1}{x^n}=x^{-n}$ (used to rewrite $\frac{1}{x}$ as $x^{-1}$)
• Simplification: $x^{-2}=\frac{1}{x^2}$

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Summary of Steps

1. Rewrite the function $y=\frac{1}{x}$ in exponent form as $y=x^{-1}$
2. Apply the power rule: bring down the exponent $(-1)$ as a coefficient, then subtract $1$ from the exponent
3. Simplify the resulting expression $(-1)x^{-2}$ to obtain the final answer $-\frac{1}{x^2}$