Finding the derivative of fractions like 1/x or 1/x² is a fundamental skill in basic calculus. You don't need the quotient rule to solve these; the Power Rule is much faster.
Rule to remember: 1/xⁿ = x⁻ⁿ
Derivative of 1/x = -1 / x²
Derivative of 1/x² = -2 / x³
Method: Convert to negative exponent, then apply Power Rule.
The Power Rule states that: d/dx [xⁿ] = n · xⁿ⁻¹ To use this, we must rewrite fractions as negative exponents.
Step 1: Rewrite 1/x as a negative exponent. 1/x = x⁻¹
Step 2: Apply the Power Rule (where n = -1). d/dx [x⁻¹] = -1 · x⁻¹⁻¹ d/dx [x⁻¹] = -1 · x⁻²
Step 3: Rewrite as a fraction. d/dx [1/x] = -1 / x²
Step 1: Rewrite 1/x² as a negative exponent. 1/x² = x⁻²
Step 2: Apply the Power Rule (where n = -2). d/dx [x⁻²] = -2 · x⁻²⁻¹ d/dx [x⁻²] = -2 · x⁻³
Step 3: Rewrite as a fraction. d/dx [1/x²] = -2 / x³
The derivative of 1/x is -1 / x². You find this by rewriting 1/x as x⁻¹ and applying the power rule.
The derivative of 1/x² is -2 / x³. This is found by rewriting it as x⁻² and applying the power rule to get -2x⁻³.
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