Find the Derivative - d/dx f(x)=1/x

Problem

$\frac{d}{dx}\frac{1}{x}$

Solution

1. Rewrite the expression using a negative exponent to prepare for the power rule.

$\frac{1}{x}=x^{-1}$

2. Apply the power rule for differentiation, which states that $\frac{d{x}^n}{dx}=n{x}^{n-1}$

$\frac{d{x}^{-1}}{dx}=-1\cdot x^{-1-1}$

3. Simplify the exponent by performing the subtraction.

$-1\cdot x^{-2}=-x^{-2}$

4. Convert the expression back into fraction form by moving the variable to the denominator.

$-x^{-2}=-\frac{1}{x^2}$

Final Answer

$\frac{d}{dx}\frac{1}{x}=-\frac{1}{x^2}$

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