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Original
2026-01-01
Modified
2026-02-28
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<p>We can derive the derivative of 5/x using proofs.</p>
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<p>We can derive the derivative of 5/x using proofs.</p>
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<p>To show this, we will use algebraic manipulation along with the rules of differentiation.</p>
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<p>To show this, we will use algebraic manipulation along with the rules of differentiation.</p>
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<p>There are several methods we use to prove this, such as:</p>
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<p>There are several methods we use to prove this, such as:</p>
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<p>By First Principle</p>
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<p>By First Principle</p>
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<p>Using Power Rule</p>
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<p>Using Power Rule</p>
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<p>Using Quotient Rule</p>
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<p>Using Quotient Rule</p>
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<p>We will now demonstrate that the differentiation of 5/x results in -5/x² using the above-mentioned methods:</p>
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<p>We will now demonstrate that the differentiation of 5/x results in -5/x² using the above-mentioned methods:</p>
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<p>By First Principle</p>
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<p>By First Principle</p>
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<p>The derivative of 5/x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
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<p>The derivative of 5/x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
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<p>To find the derivative of 5/x using the first principle, we will consider f(x) = 5/x.</p>
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<p>To find the derivative of 5/x using the first principle, we will consider f(x) = 5/x.</p>
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<p>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>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) = 5/x, we write f(x + h) = 5/(x + h).</p>
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<p>Given that f(x) = 5/x, we write f(x + h) = 5/(x + h).</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [5/(x + h) - 5/x] / h = limₕ→₀ [5x - 5(x + h)] / [h(x)(x + h)] = limₕ→₀ [-5h] / [h(x)(x + h)] = limₕ→₀ [-5] / [x(x + h)] = -5/x²</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [5/(x + h) - 5/x] / h = limₕ→₀ [5x - 5(x + h)] / [h(x)(x + h)] = limₕ→₀ [-5h] / [h(x)(x + h)] = limₕ→₀ [-5] / [x(x + h)] = -5/x²</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<p>Using Power Rule</p>
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<p>Using Power Rule</p>
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<p>To prove the differentiation of 5/x using the<a>power</a>rule, We rewrite the function as: 5/x = 5x-1.</p>
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<p>To prove the differentiation of 5/x using the<a>power</a>rule, We rewrite the function as: 5/x = 5x-1.</p>
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<p>Using the power rule, if y = xn, then dy/dx = nx(n-1).</p>
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<p>Using the power rule, if y = xn, then dy/dx = nx(n-1).</p>
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<p>So we get, d/dx (5x-1) = 5(-1)x(-1-1) = -5x-2.</p>
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<p>So we get, d/dx (5x-1) = 5(-1)x(-1-1) = -5x-2.</p>
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<p>Therefore, d/dx (5/x) = -5/x².</p>
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<p>Therefore, d/dx (5/x) = -5/x².</p>
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<p>Using Quotient Rule</p>
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<p>Using Quotient Rule</p>
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<p>We will now prove the derivative of 5/x using the quotient rule.</p>
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<p>We will now prove the derivative of 5/x using the quotient rule.</p>
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<p>The step-by-step process is demonstrated below:</p>
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<p>The step-by-step process is demonstrated below:</p>
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<p>Here, we use the formula, 5/x = 5 * (1/x).</p>
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<p>Here, we use the formula, 5/x = 5 * (1/x).</p>
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<p>Given that u = 5 and v = x⁻¹,</p>
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<p>Given that u = 5 and v = x⁻¹,</p>
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<p>Using the quotient rule formula: d/dx [u/v] = [u'v - uv']/v². u' = 0 (since u = 5 is a<a>constant</a>) v' = d/dx (x⁻¹) = -x⁻² (substitute v = x⁻¹).</p>
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<p>Using the quotient rule formula: d/dx [u/v] = [u'v - uv']/v². u' = 0 (since u = 5 is a<a>constant</a>) v' = d/dx (x⁻¹) = -x⁻² (substitute v = x⁻¹).</p>
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<p>Using the quotient rule, d/dx (5/x) = [0 * x⁻¹ - 5 * (-x⁻²)] / (x⁻¹)² = (5x⁻²)/x⁻² = -5/x².</p>
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<p>Using the quotient rule, d/dx (5/x) = [0 * x⁻¹ - 5 * (-x⁻²)] / (x⁻¹)² = (5x⁻²)/x⁻² = -5/x².</p>
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<p>Hence, d/dx (5/x) = -5/x².</p>
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<p>Hence, d/dx (5/x) = -5/x².</p>
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