# OCR A Level: Further Pure 2 - Hyperbolics [June 2008 Q7]

**Calculus**Level 3

It is given that \(f(x) = \tanh ^{ -1 }{ \left( \cfrac { 1-x }{ 2+x } \right) } \), for \(x> -\dfrac{1}{2}\)

\(\text{(i)}\) Show that \(f'(x) = -\dfrac{1}{1+2x}\), and find \(f''(x)\).

\(\text{(ii)}\) Show that the first three terms of the Maclaurin series for \(f(x)\) can be written as \(\ln a + bx + cx^2\), for some constants \(a\), \(b\) and \(c\).

**Input \(a^2+b^2+c^2\) as your answer.**