| Exam Board | Edexcel |
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| Module | C4 (Core Mathematics 4) |
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| Topic | Differentiating Transcendental Functions |
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3. (a) Given that \(a ^ { x } = \mathrm { e } ^ { k x }\), where \(a\) and \(k\) are constants, \(a > 0\) and \(x \in \mathbb { R }\), prove that \(k = \ln a\).
(b) Hence, using the derivative of \(\mathrm { e } ^ { k x }\), prove that when \(y = 2 ^ { x }\),
$$\frac { \mathrm { d } y } { \mathrm { dx } } = 2 ^ { x } \ln 2 .$$
(c) Hence deduce that the gradient of the curve with equation \(y = 2 ^ { x }\) at the point \(( 2,4 )\) is \(\ln 16\).