| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Show that derivative equals expression |
| Difficulty | Moderate -0.3 Part (i) is a straightforward application of the chain rule to a simple power function. Part (ii) requires chain rule with exponential and logarithm, but follows a standard pattern for C3 level—the 'show that' format provides the target answer, making it easier than an open 'find' question. Both parts are routine exercises testing basic chain rule competency without requiring problem-solving insight. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{\sqrt{1+2x}} \) (or \(\frac{1}{2}\cdot\frac{2}{\sqrt{1+2x}}\)) | M1 | Chain rule attempt |
| \(\frac{1}{\sqrt{1+2x}}\) | A1 | Correct answer |
| — | A1 | [3] Fully correct with working |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(y = \ln(1-e^{-x})\), \(\frac{dy}{dx} = \frac{e^{-x}}{1-e^{-x}}\) | M1 | Chain rule |
| \(= \frac{e^{-x}}{1-e^{-x}} \cdot \frac{e^x}{e^x}\) | M1 | Multiplying numerator and denominator by \(e^x\) |
| \(= \frac{1}{e^x - 1}\) | A1 | Correct simplified form |
| — | A1 | [4] Fully shown with all steps |
# Question 1:
**Part (i):**
| $\frac{1}{\sqrt{1+2x}} $ (or $\frac{1}{2}\cdot\frac{2}{\sqrt{1+2x}}$) | M1 | Chain rule attempt |
| $\frac{1}{\sqrt{1+2x}}$ | A1 | Correct answer |
| — | A1 | [3] Fully correct with working |
**Part (ii):**
| Let $y = \ln(1-e^{-x})$, $\frac{dy}{dx} = \frac{e^{-x}}{1-e^{-x}}$ | M1 | Chain rule |
| $= \frac{e^{-x}}{1-e^{-x}} \cdot \frac{e^x}{e^x}$ | M1 | Multiplying numerator and denominator by $e^x$ |
| $= \frac{1}{e^x - 1}$ | A1 | Correct simplified form |
| — | A1 | [4] Fully shown with all steps |
---1 (i) Differentiate $\sqrt { 1 + 2 x }$.\\
(ii) Show that the derivative of $\ln \left( 1 - \mathrm { e } ^ { - x } \right)$ is $\frac { 1 } { \mathrm { e } ^ { x } - 1 }$.
\hfill \mbox{\textit{OCR MEI C3 2007 Q1 [7]}}