| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find derivative of product |
| Difficulty | Moderate -0.8 This is a straightforward application of the product rule for (i) and chain rule for (ii), both standard C3 techniques requiring no problem-solving insight. The functions are simple polynomials with no algebraic manipulation needed beyond applying the rules mechanically, making this easier than average. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt use of product rule | M1 | |
| Obtain \(3x^2(x+1)^3 + 5x^3(x+1)^4\) | A1 | 2 or equiv |
| [Or: (following complete expansion and differentiation term by term) Obtain \(8x^3 + 35x^5 + 60x^4 + 50x^3 + 20x^2 + 3x^2\) | B2 | allow B1 if one term incorrect] |
| (ii) Obtain derivative of form \(kx^3(3x^4 + 1)^n\) | M1 | any constants \(k\) and \(n\) |
| Obtain derivative of form \(kx^3(3x^4 + 1)^{-1}\) | M1 | |
| Obtain correct \(6x^3(3x^4 + 1)^{-1}\) | A1 | 3 or (unsimplified) equiv |
(i) Attempt use of product rule | M1 |
Obtain $3x^2(x+1)^3 + 5x^3(x+1)^4$ | A1 | 2 or equiv
[Or: (following complete expansion and differentiation term by term) Obtain $8x^3 + 35x^5 + 60x^4 + 50x^3 + 20x^2 + 3x^2$ | B2 | allow B1 if one term incorrect]
(ii) Obtain derivative of form $kx^3(3x^4 + 1)^n$ | M1 | any constants $k$ and $n$
Obtain derivative of form $kx^3(3x^4 + 1)^{-1}$ | M1 |
Obtain correct $6x^3(3x^4 + 1)^{-1}$ | A1 | 3 or (unsimplified) equiv
---1 Differentiate each of the following with respect to $x$.\\
(i) $x ^ { 3 } ( x + 1 ) ^ { 5 }$\\
(ii) $\sqrt { 3 x ^ { 4 } + 1 }$
\hfill \mbox{\textit{OCR C3 2007 Q1 [5]}}