| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Chain rule with single composition |
| Difficulty | Moderate -0.8 This is a straightforward chain rule exercise with standard functions. Part (i) is a direct application of the chain rule to a polynomial composition, while part (ii) requires the quotient rule combined with chain rule for sin(2x). Both are routine textbook exercises requiring only mechanical application of differentiation rules with no problem-solving or insight needed. |
| Spec | 1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(u = x^2+3 \Rightarrow \frac{du}{dx} = 2x\) | M1 A1 | Chain rule, \(\frac{dy}{du}\) |
| \(y = u^5 \Rightarrow \frac{dy}{du} = 5u^4\) | A1 | |
| \(\Rightarrow \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 5u^4 \times 2x = 10x(x^2+3)^4\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Let \(u = \sin 2x \Rightarrow \frac{du}{dx} = 2\cos 2x\) | M1 | Quotient rule |
| \(v = x \Rightarrow \frac{dv}{dx} = 1\) | A1 | |
| \(\Rightarrow \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} = \frac{2x\cos 2x - \sin 2x}{x^2}\) | A1 | |
| 3 |
**(i)** $y = (x^2+3)^5$
Let $u = x^2+3 \Rightarrow \frac{du}{dx} = 2x$ | M1 A1 | Chain rule, $\frac{dy}{du}$
$y = u^5 \Rightarrow \frac{dy}{du} = 5u^4$ | A1 |
$\Rightarrow \frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx} = 5u^4 \times 2x = 10x(x^2+3)^4$ | A1 |
| 3 |
**(ii)** $y = \frac{\sin 2x}{x}$
Let $u = \sin 2x \Rightarrow \frac{du}{dx} = 2\cos 2x$ | M1 | Quotient rule
$v = x \Rightarrow \frac{dv}{dx} = 1$ | A1 |
$\Rightarrow \frac{dy}{dx} = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2} = \frac{2x\cos 2x - \sin 2x}{x^2}$ | A1 |
| 3 |3 Differentiate the following functions.\\
(i) $\quad y = \left( x ^ { 2 } + 3 \right) ^ { 5 }$\\
(ii) $y = \frac { \sin 2 x } { x }$
\hfill \mbox{\textit{OCR MEI C3 Q3 [6]}}