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Original
2026-01-01
Modified
2026-02-28
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<p>We can derive the derivative of 7x using proofs.</p>
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<p>We can derive the derivative of 7x using proofs.</p>
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<p>To show this, we will use the properties of<a>logarithms</a>and exponentials along with the rules of differentiation. There are several methods we use to prove this, such as:</p>
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<p>To show this, we will use the properties of<a>logarithms</a>and exponentials along with the rules of differentiation. There are several methods we use to prove this, such as:</p>
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<p>By Definition of Derivative</p>
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<p>By Definition of Derivative</p>
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<p>Using Logarithmic Differentiation</p>
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<p>Using Logarithmic Differentiation</p>
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<p>By Definition of Derivative</p>
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<p>By Definition of Derivative</p>
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<p>The derivative of 7x can be proved using the definition of the derivative, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of 7^x using the definition, we will consider f(x) = 7x.</p>
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<p>The derivative of 7x can be proved using the definition of the derivative, which expresses the derivative as the limit of the difference<a>quotient</a>. To find the derivative of 7^x using the definition, we will consider f(x) = 7x.</p>
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<p>Its derivative can be expressed as the following limit. f'(x) = lim_(h→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) = lim_(h→0) [f(x + h) - f(x)] / h … (1)</p>
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<p>Given that f(x) = 7x, we write f(x + h) = 7(x + h).</p>
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<p>Given that f(x) = 7x, we write f(x + h) = 7(x + h).</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = lim_(h→0) [7(x + h) - 7^x] / h = lim_(h→0) [7x * 7h - 7x] / h = 7x * lim_(h→0) [7h - 1] / h</p>
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<p>Substituting these into<a>equation</a>(1), f'(x) = lim_(h→0) [7(x + h) - 7^x] / h = lim_(h→0) [7x * 7h - 7x] / h = 7x * lim_(h→0) [7h - 1] / h</p>
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<p>We recognize this limit as the definition of the derivative of an exponential function, which results in: f'(x) = 7x ln(7)</p>
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<p>We recognize this limit as the definition of the derivative of an exponential function, which results in: f'(x) = 7x ln(7)</p>
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<p>Hence, proved.</p>
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<p>Hence, proved.</p>
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<p>Using Logarithmic Differentiation</p>
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<p>Using Logarithmic Differentiation</p>
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<p>To prove the differentiation of 7x using logarithmic differentiation,</p>
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<p>To prove the differentiation of 7x using logarithmic differentiation,</p>
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<p>Let y = 7x</p>
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<p>Let y = 7x</p>
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<p>Take the natural logarithm on both sides: ln(y) = ln(7x)</p>
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<p>Take the natural logarithm on both sides: ln(y) = ln(7x)</p>
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<p>Using logarithm properties, ln(y) = x ln(7)</p>
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<p>Using logarithm properties, ln(y) = x ln(7)</p>
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<p>Differentiate both sides with respect to x: (1/y) dy/dx = ln(7) dy/dx = y ln(7)</p>
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<p>Differentiate both sides with respect to x: (1/y) dy/dx = ln(7) dy/dx = y ln(7)</p>
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<p>Substitute y = 7x: dy/dx = 7x ln(7)</p>
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<p>Substitute y = 7x: dy/dx = 7x ln(7)</p>
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<p>Thus, we have shown that the derivative of 7x is (7x)ln(7).</p>
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<p>Thus, we have shown that the derivative of 7x is (7x)ln(7).</p>
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