0 added
0 removed
Original
2026-01-01
Modified
2026-02-28
1
<p>We can derive the derivative of 3/x using proofs. To show this, we will use the rules of differentiation.</p>
1
<p>We can derive the derivative of 3/x using proofs. To show this, we will use the rules of differentiation.</p>
2
<p>There are several methods we use to prove this, such as: By First Principle Using Power Rule Using Product Rule</p>
2
<p>There are several methods we use to prove this, such as: By First Principle Using Power Rule Using Product Rule</p>
3
<p>We will now demonstrate that the differentiation of 3/x results in -3/x² using the above-mentioned methods: By First Principle The derivative of 3/x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
3
<p>We will now demonstrate that the differentiation of 3/x results in -3/x² using the above-mentioned methods: By First Principle The derivative of 3/x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>.</p>
4
<p>To find the derivative of 3/x using the first principle, we will consider f(x) = 3/x.</p>
4
<p>To find the derivative of 3/x using the first principle, we will consider f(x) = 3/x.</p>
5
<p>Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 3/x, we write f(x + h) = 3/(x + h).</p>
5
<p>Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 3/x, we write f(x + h) = 3/(x + h).</p>
6
<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [3/(x + h) - 3/x] / h = limₕ→₀ [3x - 3(x + h)] / [x(x + h)h] = limₕ→₀ [-3h] / [x(x + h)h] = limₕ→₀ -3 / [x(x + h)] = -3 / x² Hence, the derivative of 3/x is -3/x².</p>
6
<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [3/(x + h) - 3/x] / h = limₕ→₀ [3x - 3(x + h)] / [x(x + h)h] = limₕ→₀ [-3h] / [x(x + h)h] = limₕ→₀ -3 / [x(x + h)] = -3 / x² Hence, the derivative of 3/x is -3/x².</p>
7
<p>Using Power Rule To prove the differentiation of 3/x using the<a>power</a>rule, We express 3/x as 3x⁻¹.</p>
7
<p>Using Power Rule To prove the differentiation of 3/x using the<a>power</a>rule, We express 3/x as 3x⁻¹.</p>
8
<p>Using the power rule, d/dx [x^n] = n*x^(n-1), d/dx [3x⁻¹] = -1*3x^(-1-1) = -3x⁻² = -3/x²</p>
8
<p>Using the power rule, d/dx [x^n] = n*x^(n-1), d/dx [3x⁻¹] = -1*3x^(-1-1) = -3x⁻² = -3/x²</p>
9
<p>Thus, the derivative of 3/x is -3/x².</p>
9
<p>Thus, the derivative of 3/x is -3/x².</p>
10
<p>Using Product Rule We will now prove the derivative of 3/x using the<a>product</a>rule.</p>
10
<p>Using Product Rule We will now prove the derivative of 3/x using the<a>product</a>rule.</p>
11
<p>The step-by-step process is demonstrated below: Here, we use the formula, 3/x = 3 * x⁻¹ Let u = 3 and v = x⁻¹</p>
11
<p>The step-by-step process is demonstrated below: Here, we use the formula, 3/x = 3 * x⁻¹ Let u = 3 and v = x⁻¹</p>
12
<p>Using the product rule formula: d/dx [u*v] = u'v + uv' u' = d/dx (3) = 0 v' = d/dx (x⁻¹) = -1*x⁻² d/dx (3/x) = 0 * x⁻¹ + 3 * (-1)x⁻² = -3x⁻² = -3/x²</p>
12
<p>Using the product rule formula: d/dx [u*v] = u'v + uv' u' = d/dx (3) = 0 v' = d/dx (x⁻¹) = -1*x⁻² d/dx (3/x) = 0 * x⁻¹ + 3 * (-1)x⁻² = -3x⁻² = -3/x²</p>
13
<p>Therefore, the derivative of 3/x is -3/x².</p>
13
<p>Therefore, the derivative of 3/x is -3/x².</p>
14
14