October 2020 Paper 1 Q4
4 Find the second derivative of \(\left(x^2+5\right)^4\), giving your answer in factorised form. [5]
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4\left(x^2+5\right)^3 \times 2x = 8x\left(x^2+5\right)^3\) | M1 A1 | 1.1a 1.1 |
| Using product rule with \(u = 8x \quad v = \left(x^2+5\right)^3\) | M1 | 1.1a |
| \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} = 8\left(x^2+5\right)^3 + 8x \times 3\left(x^2+5\right)^2 \times 2x\) \(= 8\left(x^2+5\right)^3 + 48x^2\left(x^2+5\right)^2\) | A1 | 1.1 |
| \(= 8\left(x^2+5\right)^2\left(7x^2+5\right)\) | A1 | 1.1 |
| [5] |
Notes
M1: Attempting to use the chain rule
A1: Any form
M1: Need not be written explicitly
FT their \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) of correct form
A1: Any form
A1: Must be factorised – allow for \(\left(x^2+5\right)^2\left(56x^2+40\right)\)
For candidates who have fully expanded the brackets and differentiate a polynomial without factorising, give SC1 for each correct term of the second derivative
\(56x^6 + 600x^4 + 1800x^2 + 1000\)