| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2017 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Standard +0.3 Part (i) is a straightforward application of the product rule with ln(3x), requiring only basic differentiation. Part (ii) uses the quotient rule and requires algebraic manipulation to reach the given form, including using the double angle identity sin(2x) = 2sin(x)cos(x). While the algebra is somewhat involved, this is a standard 'show that' question testing routine differentiation techniques with no novel problem-solving required. The question is slightly easier than average due to being guided ('show that') rather than open-ended. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07l 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 |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = 5x^2 \times \frac{3}{3x} + \ln(3x) \times 10x\) | M1 A1 | Applies Product Rule to \(y = 5x^2\ln 3x\). Expect \(\frac{dy}{dx} = Ax + Bx\ln(3x)\) for M1 where \(A\), \(B\) are positive constants |
| (2) | A1: cao - need not be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{dy}{dx} = \frac{(\sin x + \cos x)1 - x(\cos x - \sin x)}{(\sin x + \cos x)^2}\) | M1 | Applies Quotient Rule to \(y = \frac{x}{\sin x + \cos x}\). Expect \(\frac{dy}{dx} = \frac{(\sin x + \cos x)1 - x(\pm\cos x \pm \sin x)}{(\sin x + \cos x)^2}\) |
| \(\frac{dy}{dx} = \frac{(\sin x + \cos x)1 - x(\cos x - \sin x)}{(\sin^2 x + \cos^2 x) + (2\sin x\cos x)} = \frac{(\sin x + \cos x)1 - x(\cos x - \sin x)}{1 + \sin 2x}\) | B1, B1 | B1: Denominator expanded to \(\sin^2 x + \cos^2 x + \ldots\) and \((\sin^2 x + \cos^2 x) \to 1\). B1: Denominator expanded to \(\ldots + k\sin x\cos x\) and \((k\sin x\cos x) \to \frac{k}{2}\sin 2x\) |
| \(\frac{dy}{dx} = \frac{(1+x)\sin x + (1-x)\cos x}{1 + \sin 2x}\) * | A1* | cso - withheld if poor notation e.g. \(\cos x \leftrightarrow \cos\), \(\sin^2 x \leftrightarrow \sin x^2\). If only error is omission of \((\sin^2 x + \cos^2 x) \to 1\) then A1* can be awarded |
| (4) | Condone invisible brackets for M. Product rule or implicit differentiation alternatives acceptable for M1 |
# Question 6:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 5x^2 \times \frac{3}{3x} + \ln(3x) \times 10x$ | M1 A1 | Applies Product Rule to $y = 5x^2\ln 3x$. Expect $\frac{dy}{dx} = Ax + Bx\ln(3x)$ for M1 where $A$, $B$ are positive constants |
| | **(2)** | A1: cao - need not be simplified |
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = \frac{(\sin x + \cos x)1 - x(\cos x - \sin x)}{(\sin x + \cos x)^2}$ | M1 | Applies Quotient Rule to $y = \frac{x}{\sin x + \cos x}$. Expect $\frac{dy}{dx} = \frac{(\sin x + \cos x)1 - x(\pm\cos x \pm \sin x)}{(\sin x + \cos x)^2}$ |
| $\frac{dy}{dx} = \frac{(\sin x + \cos x)1 - x(\cos x - \sin x)}{(\sin^2 x + \cos^2 x) + (2\sin x\cos x)} = \frac{(\sin x + \cos x)1 - x(\cos x - \sin x)}{1 + \sin 2x}$ | B1, B1 | B1: Denominator expanded to $\sin^2 x + \cos^2 x + \ldots$ and $(\sin^2 x + \cos^2 x) \to 1$. B1: Denominator expanded to $\ldots + k\sin x\cos x$ and $(k\sin x\cos x) \to \frac{k}{2}\sin 2x$ |
| $\frac{dy}{dx} = \frac{(1+x)\sin x + (1-x)\cos x}{1 + \sin 2x}$ * | A1* | cso - withheld if poor notation e.g. $\cos x \leftrightarrow \cos$, $\sin^2 x \leftrightarrow \sin x^2$. If only error is omission of $(\sin^2 x + \cos^2 x) \to 1$ then A1* can be awarded |
| | **(4)** | Condone invisible brackets for M. Product rule or implicit differentiation alternatives acceptable for M1 |
---\begin{enumerate}
\item (i) Differentiate $y = 5 x ^ { 2 } \ln 3 x , \quad x > 0$\\
(ii) Given that
\end{enumerate}
$$y = \frac { x } { \sin x + \cos x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$
show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 1 + x ) \sin x + ( 1 - x ) \cos x } { 1 + \sin 2 x } , \quad - \frac { \pi } { 4 } < x < \frac { 3 \pi } { 4 }$$
\includegraphics[max width=\textwidth, alt={}, center]{e30f0c28-1695-40a1-8e9a-6ea7e29042bf-11_99_104_2631_1781}\\
\hfill \mbox{\textit{Edexcel C34 2017 Q6 [6]}}