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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 8^x using proofs. To show this, we will use the properties of exponential functions along with 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 8^x using proofs. To show this, we will use the properties of exponential functions along with 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 Chain Rule</li>
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<li>Using Chain 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 8^x results in 8^x ln(8) using the above-mentioned methods:</p>
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</ol><p>We will now demonstrate that the differentiation of 8^x results in 8^x ln(8) using the above-mentioned methods:</p>
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<p>By First Principle The derivative of 8^x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of 8^x using the first principle, we will consider f(x) = 8^x. Its derivative can be expressed as the following limit.</p>
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<p>By First Principle The derivative of 8^x can be proved using the First Principle, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of 8^x using the first principle, we will consider f(x) = 8^x. Its derivative can be expressed as the following limit.</p>
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<p>f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)</p>
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<p>f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1)</p>
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<p>Given that f(x) = 8^x, we write f(x + h) = 8^(x + h).</p>
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<p>Given that f(x) = 8^x, we write f(x + h) = 8^(x + h).</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [8^(x + h) - 8^x] / h = limₕ→₀ [8^x * 8^h - 8^x] / h = 8^x * limₕ→₀ [8^h - 1] / h</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = limₕ→₀ [8^(x + h) - 8^x] / h = limₕ→₀ [8^x * 8^h - 8^x] / h = 8^x * limₕ→₀ [8^h - 1] / h</p>
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<p>Using the limit property, we have limₕ→₀ [8^h - 1] / h = ln(8) Thus, f'(x) = 8^x * ln(8)</p>
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<p>Using the limit property, we have limₕ→₀ [8^h - 1] / h = ln(8) Thus, f'(x) = 8^x * ln(8)</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<p>Using Chain Rule To prove the differentiation of 8^x using the chain rule, Consider y = 8^x = e^(x ln(8)) Let u = x ln(8), then y = e^u</p>
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<p>Using Chain Rule To prove the differentiation of 8^x using the chain rule, Consider y = 8^x = e^(x ln(8)) Let u = x ln(8), then y = e^u</p>
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<p>Applying the chain rule: dy/dx = dy/du * du/dx dy/du = e^u = e^(x ln(8)) = 8^x du/dx = ln(8)</p>
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<p>Applying the chain rule: dy/dx = dy/du * du/dx dy/du = e^u = e^(x ln(8)) = 8^x du/dx = ln(8)</p>
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<p>Therefore, dy/dx = 8^x * ln(8)</p>
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<p>Therefore, dy/dx = 8^x * ln(8)</p>
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<p>Using Exponential Rule We use the standard rule for differentiating exponential functions:</p>
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<p>Using Exponential Rule We use the standard rule for differentiating exponential functions:</p>
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<p>If y = a^x, then dy/dx = a^x ln(a) Applying this to y = 8^x, we get: dy/dx = 8^x ln(8)</p>
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<p>If y = a^x, then dy/dx = a^x ln(a) Applying this to y = 8^x, we get: dy/dx = 8^x ln(8)</p>
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<p>This shows the derivative of 8^x is 8^x ln(8).</p>
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<p>This shows the derivative of 8^x is 8^x ln(8).</p>
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