June 2024 Paper 3 Q10
10 It is given that
\[\mathrm{f}(x) = 5x^3 + x\]Use differentiation from first principles to prove that
\[\mathrm{f}^{\prime}(x) = 15x^2 + 1\][5 marks]
| Scheme | Marks | AO |
|---|---|---|
| States \(5(x + h)^3 + (x + h) - \left(5x^3 + x\right)\) Condone missing brackets | M1 | 1.1a |
| Expands \((x + h)^3\) correctly Like terms need not be collected Accept all terms multiplied by 5 May be embedded | M1 | 1.1a |
| Obtains \(\dfrac{15x^2h + 15xh^2 + 5h^3 + h}{h}\) correctly eliminating \(5x^3\) and \(x\) PI by \(15x^2 + 15xh + 5h^2 + 1\) having seen a division by \(h\) Like terms need not be collected | A1 | 1.1b |
| Obtains \(15x^2 + 15xh + 5h^2 + 1\) by correctly dividing by \(h\) Like terms need not be collected | A1 | 2.1 |
| Completes a reasoned argument using the \(\displaystyle\lim_{h \to 0}\) to prove that \(\mathrm{f}^{\prime}(x) = 15x^2 + 1\) \(\mathrm{f}^{\prime}(x)\) may be seen on the final line or before Do not allow \(\dfrac{dy}{dx}\) for \(\mathrm{f}^{\prime}(x)\) | R1 | 2.5 |
| (5 marks) |