| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard differentiation rules (chain rule, product rule, quotient rule) with routine algebraic manipulation. Part (a) requires direct application of formulas, while part (b) involves showing a given form which guides the student and reduces problem-solving demand. All techniques are core C3 material with no novel insight required. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = (2x^2 - 5x + 1)^{20}\) | M1 A1 | chain rule \(20(...) f'(x)\) with no further incorrect working |
| \(\frac{dy}{dx} = 20(2x^2 - 5x + 1)^{19}(4x - 5)\) OE |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = x\cos x\) | M1 A1 | product rule \(\pm x\sin x \pm \cos x\) CSO |
| \(\frac{dy}{dx} = -x\sin x + \cos x\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \frac{x^3}{x - 2}\) | M1 | quotient rule \(\frac{±uv' ± u'v}{(x-2)^2}\) condone missing brackets |
| \(\frac{dy}{dx} = \frac{(x-2) \cdot 3x^2 - x^3 \cdot 1}{(x-2)^2}\) | A1 | |
| \(= \frac{3x^3 - 6x^2 - x^3}{(x-2)^2}\) | ||
| \(= \frac{2x^3(x - 3)}{(x-2)^2}\) | A1 | CSO |
### 1(a)(i)
$y = (2x^2 - 5x + 1)^{20}$ | M1 A1 | chain rule $20(...) f'(x)$ with no further incorrect working | **Total: 2**
$\frac{dy}{dx} = 20(2x^2 - 5x + 1)^{19}(4x - 5)$ OE |
### 1(a)(ii)
$y = x\cos x$ | M1 A1 | product rule $\pm x\sin x \pm \cos x$ CSO | **Total: 2**
$\frac{dy}{dx} = -x\sin x + \cos x$ |
### 1(b)
$y = \frac{x^3}{x - 2}$ | M1 | quotient rule $\frac{±uv' ± u'v}{(x-2)^2}$ condone missing brackets |
$\frac{dy}{dx} = \frac{(x-2) \cdot 3x^2 - x^3 \cdot 1}{(x-2)^2}$ | A1 | |
$= \frac{3x^3 - 6x^2 - x^3}{(x-2)^2}$ | | |
$= \frac{2x^3(x - 3)}{(x-2)^2}$ | A1 | CSO | **Total: 3**1
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ when:
\begin{enumerate}[label=(\roman*)]
\item $y = \left( 2 x ^ { 2 } - 5 x + 1 \right) ^ { 20 }$;
\item $y = x \cos x$.
\end{enumerate}\item Given that
$$y = \frac { x ^ { 3 } } { x - 2 }$$
show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k x ^ { 2 } ( x - 3 ) } { ( x - 2 ) ^ { 2 } }$$
where $k$ is a positive integer.
\end{enumerate}
\hfill \mbox{\textit{AQA C3 2008 Q1 [7]}}