Moderate -0.5 This is a straightforward application of the standard derivative formula for inverse tan (d/dx[tan^(-1)(u)] = 1/(1+u²) × du/dx) with chain rule. Students need to differentiate both functions, substitute x=1/2, and solve a simple equation for p. While it's a Further Maths topic, the execution is mechanical with no problem-solving required, making it easier than average.
1 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } 2 x\) and \(\mathrm { g } ( x ) = p \tan ^ { - 1 } x\), where \(p\) is a constant. Find the value of \(p\) for which \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 2 } \right) = \mathrm { g } ^ { \prime } \left( \frac { 1 } { 2 } \right)\).
1 It is given that $\mathrm { f } ( x ) = \tan ^ { - 1 } 2 x$ and $\mathrm { g } ( x ) = p \tan ^ { - 1 } x$, where $p$ is a constant. Find the value of $p$ for which $\mathrm { f } ^ { \prime } \left( \frac { 1 } { 2 } \right) = \mathrm { g } ^ { \prime } \left( \frac { 1 } { 2 } \right)$.
\hfill \mbox{\textit{OCR FP2 2010 Q1 [4]}}