| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Moderate -0.3 This is a straightforward application of the chain rule followed by the product rule. Part (i) is routine differentiation of a composite function, and part (ii) uses that result in a standard product rule calculation with algebraic simplification. The 'show that' format provides the target answer, making it slightly easier than an open-ended question. Overall, slightly below average difficulty for A-level. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = (1+3x^2)^{1/2}\) | ||
| \(dy/dx = \frac{1}{2}(1+3x^2)^{-1/2} \cdot 6x\) | M1 | chain rule |
| B1 | \(\frac{1}{2}u^{-1/2}\) | |
| \(= 3x(1+3x^2)^{-1/2}\) | A1 [3] | o.e., but must be '3'; can isw here |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = x(1+3x^2)^{1/2}\) | ||
| \(dy/dx = x \cdot \frac{3x}{\sqrt{1+3x^2}} + 1\cdot(1+3x^2)^{1/2}\) | M1 | product rule |
| A1ft | ft their \(dy/dx\) from (i) | |
| \(= \frac{3x^2+1+3x^2}{\sqrt{1+3x^2}}\) | M1 | common denominator or factoring \((1+3x^2)^{-1/2}\); must show this step for M1 E1 |
| \(= \frac{1+6x^2}{\sqrt{1+3x^2}}\) * | E1 [4] | www |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = (1+3x^2)^{1/2}$ | | |
| $dy/dx = \frac{1}{2}(1+3x^2)^{-1/2} \cdot 6x$ | M1 | chain rule |
| | B1 | $\frac{1}{2}u^{-1/2}$ |
| $= 3x(1+3x^2)^{-1/2}$ | A1 [3] | o.e., but must be '3'; can isw here |
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## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = x(1+3x^2)^{1/2}$ | | |
| $dy/dx = x \cdot \frac{3x}{\sqrt{1+3x^2}} + 1\cdot(1+3x^2)^{1/2}$ | M1 | product rule |
| | A1ft | ft their $dy/dx$ from (i) |
| $= \frac{3x^2+1+3x^2}{\sqrt{1+3x^2}}$ | M1 | common denominator or factoring $(1+3x^2)^{-1/2}$; must show this step for M1 E1 |
| $= \frac{1+6x^2}{\sqrt{1+3x^2}}$ * | E1 [4] | www |
---3 (i) Differentiate $\sqrt { 1 + 3 x ^ { 2 } }$.\\
(ii) Hence show that the derivative of $x \sqrt { 1 + 3 x ^ { 2 } }$ is $\frac { 1 + 6 x ^ { 2 } } { \sqrt { 1 + 3 x ^ { 2 } } }$.
\hfill \mbox{\textit{OCR MEI C3 2010 Q3 [7]}}