| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Hyperbolic functions |
| Type | Differentiate inverse hyperbolic functions |
| Difficulty | Standard +0.8 Part (a) is a standard inverse function differentiation using the chain rule (routine for FP3). Part (b) requires integration by parts with an inverse trig function, followed by careful algebraic manipulation and evaluation at non-trivial limits involving π/6. The 6-mark allocation and multi-step nature with exact form answer elevates this above average difficulty, but it follows standard FP3 techniques without requiring novel insight. |
| Spec | 1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08i Integration by parts |
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \arctan 3x$, and assuming the derivative of $\tan x$, prove that
$$\frac{dy}{dx} = \frac{3}{1 + 9x^2}.$$ [4]
\item Show that
$$\int_0^{\frac{\sqrt{3}}{3}} 6x \arctan 3x \, dx = \frac{1}{3}(4\pi - 3\sqrt{3}).$$ [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q5 [10]}}