| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Differentiate general exponentials |
| Difficulty | Moderate -0.8 This is a straightforward guided proof question requiring basic logarithm manipulation and application of the chain rule to exponential functions. All steps are explicitly prompted with 'hence', making it easier than average as students simply follow the scaffolded structure rather than devise their own approach. |
| Spec | 1.07j Differentiate exponentials: e^(kx) and a^(kx) |
\begin{enumerate}[label=(\alph*)]
\item Given that $a^x = e^{kx}$, where $a$ and $k$ are constants, $a > 0$ and $x \in \mathbb{R}$, prove that $k = \ln a$. [2]
\item Hence, using the derivative of $e^{kx}$, prove that when $y = 2^x$,
$$\frac{dy}{dx} = 2^x \ln 2.$$ [2]
\item Hence deduce that the gradient of the curve with equation $y = 2^x$ at the point $(2, 4)$ is $\ln 16$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [6]}}