| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of quotient |
| Difficulty | Moderate -0.3 This is a straightforward multi-part differentiation question testing standard product rule, quotient rule, and implicit differentiation techniques. Part (a)(i) is routine product rule with basic functions, (a)(ii) requires quotient rule and algebraic simplification, and part (b) involves chain rule with inverse trig. All are textbook-standard applications with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \(\frac{d}{dx}(\ln(3x)) = \frac{3}{3x}\) | M1 | Attempts to differentiate \(\ln(3x)\) to \(\frac{B}{x}\). Note that \(\frac{1}{3x}\) is fine. |
| \(\frac{d}{dx}(x^2 \ln(3x)) = \ln(3x) \times \frac{1}{2} x^{-\frac{1}{2}} + x^2 \times \frac{1}{3x}\) | M1A1 | Attempts the product rule for \(x^{\frac{1}{2}} \ln(3x)\). If the rule is quoted it must be correct. There must have been some attempt to differentiate both terms. If the rule is not quoted nor implied from their stating of |
| (a)(i) $\frac{d}{dx}(\ln(3x)) = \frac{3}{3x}$ | M1 | Attempts to differentiate $\ln(3x)$ to $\frac{B}{x}$. Note that $\frac{1}{3x}$ is fine. |
|---|---|---|
| $\frac{d}{dx}(x^2 \ln(3x)) = \ln(3x) \times \frac{1}{2} x^{-\frac{1}{2}} + x^2 \times \frac{1}{3x}$ | M1A1 | Attempts the product rule for $x^{\frac{1}{2}} \ln(3x)$. If the rule is quoted it must be correct. There must have been some attempt to differentiate both terms. If the rule is not quoted nor implied from their stating of\begin{enumerate}[label=(\alph*)]
\item Differentiate with respect to $x$,
\begin{enumerate}[label=(\roman*)]
\item $x ^ { \frac { 1 } { 2 } } \ln ( 3 x )$
\item $\frac { 1 - 10 x } { ( 2 x - 1 ) ^ { 5 } }$, giving your answer in its simplest form.
\end{enumerate}\item Given that $x = 3 \tan 2 y$ find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $x$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2012 Q7 [11]}}