| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Moderate -0.8 This is a straightforward multi-part differentiation question testing standard rules. Parts (a) and (b) are routine chain rule applications, while part (c) combines them using the product rule. All techniques are direct applications with no problem-solving required, making it easier than average but not trivial due to the algebraic manipulation needed in part (c). |
| 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 rates |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 5(3x+1)^4 \times 3 = 15(3x+1)^4\) | M1, A1 | \(k(3x+1)^4\) with no further errors (w.n.f.e) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = \frac{3}{3x+1}\) | M1, A1 | \(\frac{k}{3x+1}\) w.n.f.e |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = (3x+1)^2 \times \frac{3}{3x+1} + \ln(3x+1) \times 15(3x+1)^4 = (3x+1)^4[3 + 15\ln(3x+1)] = 3(3x+1)^4[1 + 5\ln(3x+1)]\) | M1, A1, A1 | Product rule \(uv' + u'v\) (from (a) and (b)); either term correct; CSO with no further errors |
**1(a)**
| $\frac{dy}{dx} = 5(3x+1)^4 \times 3 = 15(3x+1)^4$ | M1, A1 | $k(3x+1)^4$ with no further errors (w.n.f.e) |
**1(b)**
| $\frac{dy}{dx} = \frac{3}{3x+1}$ | M1, A1 | $\frac{k}{3x+1}$ w.n.f.e |
**1(c)**
| $\frac{dy}{dx} = (3x+1)^2 \times \frac{3}{3x+1} + \ln(3x+1) \times 15(3x+1)^4 = (3x+1)^4[3 + 15\ln(3x+1)] = 3(3x+1)^4[1 + 5\ln(3x+1)]$ | M1, A1, A1 | Product rule $uv' + u'v$ (from (a) and (b)); either term correct; CSO with no further errors |
**Total: 7 marks**
---1 Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when:
\begin{enumerate}[label=(\alph*)]
\item $y = ( 3 x + 1 ) ^ { 5 }$;
\item $y = \ln ( 3 x + 1 )$;
\item $y = ( 3 x + 1 ) ^ { 5 } \ln ( 3 x + 1 )$.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2008 Q1 [7]}}