| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find tangent line equation |
| Difficulty | Moderate -0.3 This is a structured, multi-part question that guides students through deriving the derivative of logarithms to any base, then applies it to find a tangent line. Parts (a) and (b) involve standard logarithm manipulation and implicit differentiation that C3 students should know. Parts (c) and (d) are routine applications requiring substitution into y=mx+c and solving for x-intercept. While it has multiple parts (10 marks total), each step is straightforward with clear scaffolding, making it slightly easier than a typical C3 question. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations |
Given that $y = \log_a x$, $x > 0$, where $a$ is a positive constant,
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item express $x$ in terms of $a$ and $y$, [1]
\item deduce that $\ln x = y \ln a$. [1]
\end{enumerate}
\item Show that $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{x \ln a}$. [2]
\end{enumerate}
The curve $C$ has equation $y = \log_{10} x$, $x > 0$. The point $A$ on $C$ has $x$-coordinate 10. Using the result in part (b),
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find an equation for the tangent to $C$ at $A$. [4]
\end{enumerate}
The tangent to $C$ at $A$ crosses the $x$-axis at the point $B$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the exact $x$-coordinate of $B$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q7 [10]}}