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2026-01-01
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2026-02-28
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<p>We can derive the derivative of 5e^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 5e^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 Constant Multiple Rule</li>
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<li>Using Constant Multiple Rule</li>
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<li>Using Exponential Rule</li>
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<li>Using Exponential Rule</li>
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</ol><p>We will now demonstrate that the differentiation of 5e^x results in 5e^x using the above-mentioned methods:</p>
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</ol><p>We will now demonstrate that the differentiation of 5e^x results in 5e^x 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 5e^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 5e^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 5e^x using the first principle, we will consider f(x) = 5e^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 5e^x using the first principle, we will consider f(x) = 5e^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) = 5e^x, we write f(x + h) = 5e^(x + h).</p>
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<p>Given that f(x) = 5e^x, we write f(x + h) = 5e^(x + h).</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [5e^(x + h) - 5e^x] / h = 5 limₕ→₀ [e^x (e^h - 1)] / h</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [5e^(x + h) - 5e^x] / h = 5 limₕ→₀ [e^x (e^h - 1)] / h</p>
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<p>We know that e^h - 1 is approximately h for small h, f'(x) = 5e^x limₕ→₀ h / h = 5e^x</p>
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<p>We know that e^h - 1 is approximately h for small h, f'(x) = 5e^x limₕ→₀ h / h = 5e^x</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<h3>Using Constant Multiple Rule</h3>
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<h3>Using Constant Multiple Rule</h3>
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<p>To prove the differentiation of 5e^x using the constant<a>multiple</a>rule, We use the formula: d/dx (cf(x)) = c d/dx f(x) Let c = 5 and f(x) = e^x. d/dx (5e^x) = 5 d/dx (e^x)</p>
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<p>To prove the differentiation of 5e^x using the constant<a>multiple</a>rule, We use the formula: d/dx (cf(x)) = c d/dx f(x) Let c = 5 and f(x) = e^x. d/dx (5e^x) = 5 d/dx (e^x)</p>
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<p>Since the derivative of e^x is e^x, = 5e^x</p>
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<p>Since the derivative of e^x is e^x, = 5e^x</p>
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<h3>Using Exponential Rule</h3>
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<h3>Using Exponential Rule</h3>
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<p>The exponential rule states that the derivative of e^x is e^x. For the function 5e^x, the derivative is: d/dx (5e^x) = 5 d/dx (e^x) = 5e^x</p>
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<p>The exponential rule states that the derivative of e^x is e^x. For the function 5e^x, the derivative is: d/dx (5e^x) = 5 d/dx (e^x) = 5e^x</p>
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<p>Thus, the derivative of 5e^x is 5e^x.</p>
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<p>Thus, the derivative of 5e^x is 5e^x.</p>
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