| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Second derivative and nature determination |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring chain rule application to arctan(√x), followed by algebraic verification of a second derivative identity. Part (a) is straightforward differentiation and substitution. Part (b) requires finding d²y/dx² and algebraic manipulation to verify the given relation—more demanding than standard C3/C4 but routine for FP3 students familiar with implicit differentiation techniques. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates4.08g Derivatives: inverse trig and hyperbolic functions |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = \frac{1}{1+x} - \frac{1}{2}x^{-1} = \left(\frac{1}{2x^2(1+x)}\right)\) | M1 A1 |
| \(x = \frac{1}{4} \Rightarrow \frac{dy}{dx} = \frac{4}{5}\) | A1 (3) |
| Answer | Marks |
|---|---|
| \(\frac{dy}{dx} = \frac{1}{2}(1+x)^{-1}x^{-\frac{1}{2}}\) | |
| \(\frac{d^2y}{dx^2} = -\frac{1}{2}(1+x)^{-2} \times x^{-\frac{1}{2}} - \frac{1}{4}(1+x)^{-1}x^{-\frac{3}{2}}\) | M1 A1 |
| \(= -\frac{1+3x}{4x^{\frac{3}{2}}(1+x)^2}\) | |
| \(2x(1+x)\frac{d^2y}{dx^2} + (1+3x)\frac{dy}{dx}\) | |
| \(= 2x(1+x)\left(-\frac{1+3x}{4x^{\frac{3}{2}}(1+x)^2}\right) + (1+3x)\left(\frac{1}{2x^{\frac{1}{2}}(1+x)}\right)\) | M1 A1, A1 |
| \(= 0\) ⭐ | A1 cso (6) |
**(a)**
$\frac{dy}{dx} = \frac{1}{1+x} - \frac{1}{2}x^{-1} = \left(\frac{1}{2x^2(1+x)}\right)$ | M1 A1 |
$x = \frac{1}{4} \Rightarrow \frac{dy}{dx} = \frac{4}{5}$ | A1 (3) |
**(b)**
$\frac{dy}{dx} = \frac{1}{2}(1+x)^{-1}x^{-\frac{1}{2}}$ | |
$\frac{d^2y}{dx^2} = -\frac{1}{2}(1+x)^{-2} \times x^{-\frac{1}{2}} - \frac{1}{4}(1+x)^{-1}x^{-\frac{3}{2}}$ | M1 A1 |
$= -\frac{1+3x}{4x^{\frac{3}{2}}(1+x)^2}$ | |
$2x(1+x)\frac{d^2y}{dx^2} + (1+3x)\frac{dy}{dx}$ | |
$= 2x(1+x)\left(-\frac{1+3x}{4x^{\frac{3}{2}}(1+x)^2}\right) + (1+3x)\left(\frac{1}{2x^{\frac{1}{2}}(1+x)}\right)$ | M1 A1, A1 |
$= 0$ ⭐ | A1 cso (6) |
(9 marks)4. $y = \arctan ( \sqrt { } x ) , \quad x > 0,0 < y < \frac { \pi } { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find the value of $\frac { \mathrm { dy } } { \mathrm { dx } }$ at $\mathrm { x } = \frac { 1 } { 4 }$.
\item Show that $2 x ( 1 + x ) \frac { d ^ { 2 } y } { d x ^ { 2 } } + ( 1 + 3 x ) \frac { d y } { d x } = 0$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q4 [9]}}