| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Basic power rule differentiation |
| Difficulty | Moderate -0.8 This is a straightforward differentiation exercise requiring only basic power rule manipulation (rewriting surds as fractional powers) and algebraic verification. All three parts are routine calculations with no problem-solving insight needed, making it easier than average for A-level, though the verification in part (c) adds minor algebraic work. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}} + 2x^{-\frac{1}{2}}\) | M1 A2 | |
| (b) \(\frac{d^2y}{dx^2} = -\frac{1}{4}x^{-\frac{3}{2}} - 3x^{-\frac{3}{2}}\) | M1 A1 | |
| (c) LHS \(= 4x\left(-\frac{1}{4}x^{-\frac{3}{2}} - 3x^{-\frac{3}{2}}\right) + 4x\left(\frac{1}{2}x^{-\frac{1}{2}} + 2x^{-\frac{1}{2}}\right) - \left(x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\right)\) | M1 A1 | |
| \(= -x^{\frac{1}{2}} - 12x^{-\frac{1}{2}} + 2x^{\frac{1}{2}} + 8x^{-\frac{1}{2}} - x^{\frac{1}{2}} + 4x^{-\frac{1}{2}}\) | ||
| \(= 0\) | A1 | (8 marks) |
**(a)** $\frac{dy}{dx} = \frac{1}{2}x^{-\frac{1}{2}} + 2x^{-\frac{1}{2}}$ | M1 A2 |
**(b)** $\frac{d^2y}{dx^2} = -\frac{1}{4}x^{-\frac{3}{2}} - 3x^{-\frac{3}{2}}$ | M1 A1 |
**(c)** LHS $= 4x\left(-\frac{1}{4}x^{-\frac{3}{2}} - 3x^{-\frac{3}{2}}\right) + 4x\left(\frac{1}{2}x^{-\frac{1}{2}} + 2x^{-\frac{1}{2}}\right) - \left(x^{\frac{1}{2}} - 4x^{-\frac{1}{2}}\right)$ | M1 A1 |
$= -x^{\frac{1}{2}} - 12x^{-\frac{1}{2}} + 2x^{\frac{1}{2}} + 8x^{-\frac{1}{2}} - x^{\frac{1}{2}} + 4x^{-\frac{1}{2}}$ | |
$= 0$ | A1 | (8 marks)Given that
$$y = \sqrt{x} - \frac{4}{\sqrt{x}},$$
\begin{enumerate}[label=(\alph*)]
\item find $\frac{dy}{dx}$. [3]
\item find $\frac{d^2y}{dx^2}$. [2]
\item show that
$$4x\frac{d^2y}{dx^2} + 4x\frac{dy}{dx} - y = 0.$$ [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q7 [8]}}