C3 January 2012 Q1
1. Differentiate with respect to \(x\), giving your answer in its simplest form,
| Scheme | Marks |
|---|---|
| \(\dfrac{d}{dx}(\ln(3x)) \to \dfrac{B}{x}\) for any constant \(B\) | M1 |
| Applying vu’+uv’, \(\ln(3x) \times 2x + x\) | M1, A1 A1 |
| (4) |
Notes
M1 Differentiates the \(\ln(3x)\) term to \(\frac{B}{x}\). Note that \(\frac{1}{3x}\) is fine for this mark.
M1 Applies the product rule to \(x^2\ln(3x)\). If the rule is quoted it must be correct.
There must have been some attempt to differentiate both terms.
If the rule is not quoted (or implied by their working) only accept answers of the form
\(\ln(3x) \times Ax + x^2 \times \frac{B}{x}\) where A and B are non- zero constants
A1 One term correct and simplified, either \(2x\ln(3x)\) or \(x\). \(\ln 3x^{2x}\) and \(\ln(3x)\,2x\) are acceptable forms
A1 Both terms correct and simplified on the same line. \(2x\ln(3x) + x\), \(\ln(3x) \times 2x + x\), \(x(2\ln 3x + 1)\) oe
| Scheme | Marks |
|---|---|
| Applying \(\dfrac{vu' - uv'}{v^2}\) | |
| \(\dfrac{x^3 \times 4\cos(4x) - \sin(4x) \times 3x^2}{x^6}\) | M1 A1+A1 A1 |
| \(= \dfrac{4x\cos(4x) - 3\sin(4x)}{x^4}\) | A1 |
| (5) | |
| (9 marks) |
Notes
M1 Applies the quotient rule. A version of this appears in the formula booklet. If the formula is quoted it must be correct. There must have been some attempt to differentiate both terms.
If the formula is not quoted (nor implied by their working) only accept answers of the form
\(\dfrac{x^3 \times \pm A\cos(4x) - \sin(4x) \times Bx^2}{(x^3)^2 \text{ or } x^6 \text{ or } x^5 \text{ or } x^9}\) with \(B \gt 0\)
A1 Correct first term on numerator \(x^3 \times 4\cos(4x)\)
A1 Correct second term on numerator \(-\sin(4x) \times 3x^2\)
A1 Correct denominator \(x^6\), the \((x^3)^2\) needs to be simplified
A1 Fully correct simplified expression \(\dfrac{4x\cos(4x) - 3\sin(4x)}{x^4}\), \(\dfrac{\cos(4x)4x - \sin(4x)3}{x^4}\) oe.
Accept \(4x^{-3}\cos(4x) - 3x^{-4}\sin(4x)\) oe
Alternative method using the product rule.
M1,A1 Writes \(\dfrac{\sin(4x)}{x^3}\) as \(\sin(4x) \times x^{-3}\) and applies the product rule. They will score both of these marks or neither of them. If the formula is quoted it must be correct. There must have been some attempt to differentiate both terms. If the formula is not quoted (nor implied by their working) only accept answers of the form \(x^{-3} \times A\cos(4x) + \sin(4x) \times \pm Bx^{-4}\)
A1 One term correct, either \(x^{-3} \times 4\cos(4x)\) or \(\sin(4x) \times -3x^{-4}\)
A1 Both terms correct,Eg. \(x^{-3} \times 4\cos(4x) + \sin(4x) \times -3x^{-4}\).
A1 Fully correct expression. \(4x^{-3}\cos(4x) - 3x^{-4}\sin(4x)\) or \(4\cos(4x)x^{-3} - 3\sin(4x)x^{-4}\) oe
The negative must have been dealt with for the final mark.