| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2015 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Moderate -0.8 Part (a) is a straightforward application of the product rule with simple polynomial functions, requiring only mechanical execution. Part (b) is basic differentiation using the power rule followed by substitution. Both are routine C1 exercises with no problem-solving required, making this easier than average but not trivial since it does require correct application of standard techniques. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((x^2+3)(5-x)=5x^2-x^3+15-3x\) | M1 | Attempt to multiply out brackets, must have four terms, at least three correct |
| A1 | Fully correct expression. Do not ISW if signs then changed. Max 2/4 | |
| \(\frac{dy}{dx}=10x-3x^2-3\) | M1 | Attempt to differentiate their expression (power of at least one term involving \(x\) reduced by one) |
| A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dy}{dx}=-\frac{1}{3}x^{-\frac{4}{3}}\) | M1 | Attempt to differentiate i.e. \(-\frac{1}{3}x^{-\frac{k}{3}}\) soi for positive integer \(k\). Note: \(x^{-\frac{1}{3}}\) misread as \(x^{\frac{1}{3}}\) earns max 2/4: \(\frac{dy}{dx}=\frac{1}{3}x^{-\frac{2}{3}}\) M1 A0 MR |
| A1 | Fully correct | |
| When \(x=-8\), \(\frac{dy}{dx}=-\frac{1}{3}\times(-8)^{-\frac{4}{3}}\) | B1 | \((-8)^{-\frac{4}{3}}=\frac{1}{16}\) www. Must use \(-8\). Note: \((-8)^{-\frac{2}{3}}=\frac{1}{4}\) B1 |
| \(\frac{dy}{dx}=-\frac{1}{3}\times\frac{1}{16}=-\frac{1}{48}\) | A1 | Final answer. Note: Final answer \(\frac{1}{12}\) A0 MR |
## Question 7:
### Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(x^2+3)(5-x)=5x^2-x^3+15-3x$ | M1 | Attempt to multiply out brackets, must have four terms, at least three correct |
| | A1 | Fully correct expression. Do not ISW if signs then changed. Max 2/4 |
| $\frac{dy}{dx}=10x-3x^2-3$ | M1 | Attempt to differentiate their expression (power of at least one term involving $x$ reduced by one) |
| | A1 | |
**Alternative using product rule:** Clear attempt at correct rule M1*; Both expressions fully correct A1; Expand brackets of both parts M1*dep; Fully correct expression A1
### Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx}=-\frac{1}{3}x^{-\frac{4}{3}}$ | M1 | Attempt to differentiate i.e. $-\frac{1}{3}x^{-\frac{k}{3}}$ soi for positive integer $k$. Note: $x^{-\frac{1}{3}}$ misread as $x^{\frac{1}{3}}$ earns max 2/4: $\frac{dy}{dx}=\frac{1}{3}x^{-\frac{2}{3}}$ M1 A0 MR |
| | A1 | Fully correct |
| When $x=-8$, $\frac{dy}{dx}=-\frac{1}{3}\times(-8)^{-\frac{4}{3}}$ | B1 | $(-8)^{-\frac{4}{3}}=\frac{1}{16}$ www. Must use $-8$. Note: $(-8)^{-\frac{2}{3}}=\frac{1}{4}$ B1 |
| $\frac{dy}{dx}=-\frac{1}{3}\times\frac{1}{16}=-\frac{1}{48}$ | A1 | Final answer. Note: Final answer $\frac{1}{12}$ A0 MR |
---7
\begin{enumerate}[label=(\alph*)]
\item Given that $\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 \right) ( 5 - x )$, find $\mathrm { f } ^ { \prime } ( x )$.
\item Find the gradient of the curve $y = x ^ { - \frac { 1 } { 3 } }$ at the point where $x = - 8$.
\end{enumerate}
\hfill \mbox{\textit{OCR C1 2015 Q7 [8]}}