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Original
2026-01-01
Modified
2026-02-28
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<p>We can derive the derivative of 1/x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:</p>
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<p>We can derive the derivative of 1/x using proofs. To show this, we will use the rules of differentiation. There are several methods we use to prove this, such as:</p>
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<ol><li>By First Principle</li>
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<ol><li>By First Principle</li>
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<li>Using Power Rule</li>
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<li>Using Power Rule</li>
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<li>Using Quotient Rule</li>
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<li>Using Quotient Rule</li>
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</ol><p>We will now demonstrate that the differentiation of 1/x results in -1/x² using the above-mentioned methods:</p>
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</ol><p>We will now demonstrate that the differentiation of 1/x results in -1/x² using the above-mentioned methods:</p>
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<h3>By First Principle</h3>
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<h3>By First Principle</h3>
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<p>The derivative of 1/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.</p>
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<p>The derivative of 1/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.</p>
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<p>To find the derivative of 1/x using the first principle, we will consider f(x) = 1/x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)</p>
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<p>To find the derivative of 1/x using the first principle, we will consider f(x) = 1/x. Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)</p>
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<p>Given that f(x) = 1/x, we write f(x + h) = 1/(x + h).</p>
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<p>Given that f(x) = 1/x, we write f(x + h) = 1/(x + h).</p>
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<p>Substituting these into<a>equation</a>(1),</p>
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<p>Substituting these into<a>equation</a>(1),</p>
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<p>f'(x) = limₕ→₀ [1/(x + h) - 1/x] / h = limₕ→₀ [(x - (x + h)) / (x(x + h))] / h = limₕ→₀ [-h / (x² + xh)] / h = limₕ→₀ -1 / (x² + xh) = -1/x²</p>
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<p>f'(x) = limₕ→₀ [1/(x + h) - 1/x] / h = limₕ→₀ [(x - (x + h)) / (x(x + h))] / h = limₕ→₀ [-h / (x² + xh)] / h = limₕ→₀ -1 / (x² + xh) = -1/x²</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<h3>Using Power Rule</h3>
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<h3>Using Power Rule</h3>
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<p>To prove the differentiation of 1/x using the<a>power</a>rule, We rewrite 1/x as x⁻¹.</p>
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<p>To prove the differentiation of 1/x using the<a>power</a>rule, We rewrite 1/x as x⁻¹.</p>
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<p>By the power rule: d/dx [xⁿ] = nxⁿ⁻¹ Let n = -1, d/dx (x⁻¹) = -1 * x⁻² = -1/x²</p>
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<p>By the power rule: d/dx [xⁿ] = nxⁿ⁻¹ Let n = -1, d/dx (x⁻¹) = -1 * x⁻² = -1/x²</p>
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<h3>Using Quotient Rule</h3>
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<h3>Using Quotient Rule</h3>
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<p>We will now prove the derivative of 1/x using the quotient rule. The step-by-step process is demonstrated below:</p>
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<p>We will now prove the derivative of 1/x using the quotient rule. The step-by-step process is demonstrated below:</p>
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<p>Here, 1/x can be viewed as (1)/(x)</p>
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<p>Here, 1/x can be viewed as (1)/(x)</p>
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<p>Using the quotient rule formula:</p>
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<p>Using the quotient rule formula:</p>
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<p>d/dx [u/v] = (v * u' - u * v') / v²</p>
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<p>d/dx [u/v] = (v * u' - u * v') / v²</p>
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<p>Let u = 1 (constant), v = x u' = 0, v' = 1</p>
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<p>Let u = 1 (constant), v = x u' = 0, v' = 1</p>
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<p>d/dx (1/x) = (x * 0 - 1 * 1) / x² = -1/x²</p>
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<p>d/dx (1/x) = (x * 0 - 1 * 1) / x² = -1/x²</p>
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<p>Thus, d/dx (1/x) = -1/x².</p>
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<p>Thus, d/dx (1/x) = -1/x².</p>
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