| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Moderate -0.3 This is a straightforward application of the product rule followed by routine tangent line work. Part (a) requires standard product rule differentiation of x³ln(x). Parts (b)(i) and (b)(ii) involve substituting x=e, finding the tangent equation, and solving for an x-intercept—all mechanical steps with no conceptual challenges. Slightly easier than average due to the clean substitution at x=e. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07m Tangents and normals: gradient and equations1.07q Product and quotient rules: differentiation |
3 A curve has equation $y = x ^ { 3 } \ln x$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item \begin{enumerate}[label=(\roman*)]
\item Find an equation of the tangent to the curve $y = x ^ { 3 } \ln x$ at the point on the curve where $x = \mathrm { e }$.
\item This tangent intersects the $x$-axis at the point $A$. Find the exact value of the $x$-coordinate of the point $A$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2012 Q3 [7]}}