| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find derivative after algebraic simplification (fractional/mixed powers) |
| Difficulty | Easy -1.3 This is a straightforward C1 differentiation question requiring basic application of power rule. Part (a) needs rewriting √x as x^(1/2) and applying standard rules. Part (b) requires expanding the numerator and simplifying before differentiating—routine algebraic manipulation with no problem-solving insight needed. |
| Spec | 1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (a) \(y = x^4 + 6x^{\frac{1}{2}} \Rightarrow y' = 4x^3 + 3x^{-\frac{1}{2}}\) or \(4x^3 + \frac{3}{\sqrt{x}}\) | M1A1A1 | M1: attempt to differentiate \(x^n \to x^{n-1}\); 1st A1: one correct term; 2nd A1: both correct. Note \(4x^3 + 3x^{-\frac{1}{2}} + c\) scores M1A1A0 |
| (b) \((x+4)^2 = x^2 + 8x + 16\) | M1 | Must have \(x^2, x, x^0\) terms with at least 2 correct, e.g. \(x^2+8x+8\) or \(x^2+2x+16\) |
| \(\frac{(x+4)^2}{x} = x + 8 + 16x^{-1}\) (allow 4+4 for 8) | A1 | Correct expression. Allow \(\frac{16}{x}\) and \(8x^0\) |
| \(y' = 1 - 16x^{-2}\) | M1A1 | 2nd M1: any correct differentiation. Note \(\frac{x^2+8x+16}{x}\) giving \((2x+8)/1\) is M0A0 |
## Question 5:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(a)** $y = x^4 + 6x^{\frac{1}{2}} \Rightarrow y' = 4x^3 + 3x^{-\frac{1}{2}}$ or $4x^3 + \frac{3}{\sqrt{x}}$ | M1A1A1 | M1: attempt to differentiate $x^n \to x^{n-1}$; 1st A1: one correct term; 2nd A1: both correct. Note $4x^3 + 3x^{-\frac{1}{2}} + c$ scores M1A1A0 |
| **(b)** $(x+4)^2 = x^2 + 8x + 16$ | M1 | Must have $x^2, x, x^0$ terms with at least 2 correct, e.g. $x^2+8x+8$ or $x^2+2x+16$ |
| $\frac{(x+4)^2}{x} = x + 8 + 16x^{-1}$ (allow 4+4 for 8) | A1 | Correct expression. Allow $\frac{16}{x}$ and $8x^0$ |
| $y' = 1 - 16x^{-2}$ | M1A1 | 2nd M1: any correct differentiation. Note $\frac{x^2+8x+16}{x}$ giving $(2x+8)/1$ is M0A0 |5. Differentiate with respect to $x$
\begin{enumerate}[label=(\alph*)]
\item $x ^ { 4 } + 6 \sqrt { } x$,
\item $\frac { ( x + 4 ) ^ { 2 } } { x }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 2006 Q5 [7]}}