| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2023 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Second derivative and nature determination |
| Difficulty | Easy -1.3 This is a straightforward differentiation exercise requiring only basic power rule application (no chain rule despite the topic label). Students rewrite 3/x² as 3x⁻², differentiate term-by-term twice, and simplify. It's purely procedural with no problem-solving element, making it easier than average A-level questions. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Scheme | Marks | Guidance |
| \(\frac{dy}{dx} = 15x^2 - 6x^{-3} - 7\) | M1A1A1 | M1: Decreases power by 1 on at least one term. A1: Two correct unsimplified terms (indices must be processed). A1: \(15x^2 - 6x^{-3} - 7\) or simplified equivalent. Condone \(7x^0\) as unsimplified. Do not accept \(15x^2 - 6x^{-3} - 7x^0\). |
| \(\frac{d^2y}{dx^2} = 30x + 18x^{-4}\) | M1A1 | M1: Decreases power by 1 on one of the terms of their changed function. Do not allow this mark for constant of integration in (a) → 0 in (b). A1: \(30x + 18x^{-4}\) or simplified equivalent e.g. \(30x + \frac{18}{x^4}\). Isw once correct answer seen. Withhold this mark if constant of integration present and not already penalised in (a). Allow mark following \(15x^2 - 6x^{-3} \pm 7\) in (a). |
| Answer/Scheme | Marks | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 15x^2 - 6x^{-3} - 7$ | M1A1A1 | M1: Decreases power by 1 on at least one term. A1: Two correct unsimplified terms (indices must be processed). A1: $15x^2 - 6x^{-3} - 7$ or simplified equivalent. Condone $7x^0$ as unsimplified. Do not accept $15x^2 - 6x^{-3} - 7x^0$. |
| $\frac{d^2y}{dx^2} = 30x + 18x^{-4}$ | M1A1 | M1: Decreases power by 1 on one of the terms of their changed function. Do not allow this mark for constant of integration in (a) → 0 in (b). A1: $30x + 18x^{-4}$ or simplified equivalent e.g. $30x + \frac{18}{x^4}$. Isw once correct answer seen. Withhold this mark if constant of integration present and not already penalised in (a). Allow mark following $15x^2 - 6x^{-3} \pm 7$ in (a). |
**Total: 5 marks**
---\begin{enumerate}
\item Given that
\end{enumerate}
$$y = 5 x ^ { 3 } + \frac { 3 } { x ^ { 2 } } - 7 x \quad x > 0$$
find, in simplest form,\\
(a) $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\
(b) $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
\hfill \mbox{\textit{Edexcel P1 2023 Q1 [5]}}