| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiation from First Principles |
| Type | First principles: x³ terms |
| Difficulty | Moderate -0.5 This is a standard first principles differentiation question for a single power term (x³). While it requires knowledge of the binomial expansion and algebraic manipulation, it's a routine textbook exercise that follows a well-established method. It's slightly easier than average because it's a direct application of a standard technique with no problem-solving required, though not trivial since students must execute the proof correctly. |
| Spec | 1.07g Differentiation from first principles: for small positive integer powers of x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Considers \(\frac{(x+h)^3 - x^3}{h}\) | B1 | 2.1 — Correct fraction for gradient of chord. May also be awarded for expanded form. Does not have to be linked to gradient of chord |
| Expands \((x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\) | M1 | 1.1b — Look for two correct terms, most likely \(x^3 + ... + h^3\). Independent of B1 |
| Gradient of chord \(= \frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2\) | A1 | 1.1b — Exact unsimplified equivalent such as \(3x^2 + 2xh + xh + h^2\) also acceptable |
| As \(h \to 0\), \(3x^2 + 3xh + h^2 \to 3x^2\) so derivative \(= 3x^2\) | A1* | 2.5 — CSO. Must link gradient of chord to gradient of curve. Do not allow \(h=0\) alone without limit being considered. Condone invisible brackets for expansion of \((x+h)^3\) as long as only seen as intermediate working |
## Question 10:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Considers $\frac{(x+h)^3 - x^3}{h}$ | B1 | 2.1 — Correct fraction for gradient of chord. May also be awarded for expanded form. Does not have to be linked to gradient of chord |
| Expands $(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3$ | M1 | 1.1b — Look for two correct terms, most likely $x^3 + ... + h^3$. Independent of B1 |
| Gradient of chord $= \frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2$ | A1 | 1.1b — Exact unsimplified equivalent such as $3x^2 + 2xh + xh + h^2$ also acceptable |
| As $h \to 0$, $3x^2 + 3xh + h^2 \to 3x^2$ so derivative $= 3x^2$ | A1* | 2.5 — CSO. Must link gradient of chord to gradient of curve. Do not allow $h=0$ alone without limit being considered. Condone invisible brackets for expansion of $(x+h)^3$ as long as only seen as intermediate working |\begin{enumerate}
\item Prove, from first principles, that the derivative of $x ^ { 3 }$ is $3 x ^ { 2 }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 2018 Q10 [4]}}