Challenging +1.8 This question requires differentiation from first principles of a complex composite function involving a square root of a quotient, demanding sophisticated algebraic manipulation (rationalizing, simplifying nested radicals) and multi-step reasoning. While the technique is standard for Further Maths, the algebraic complexity and the need to reach a specific non-obvious form elevates this significantly above routine A-level questions.
2. A function is defined by:
$$f ( x ) = \sqrt { \frac { 1 - x } { 1 + x } } , x \in \mathbb { R } , | x | < 1$$
a) P and Q are points on the curve with \(x\)-coordinates \(x\) and \(x + h\) respectively. Find the gradient of the line segment PQ . Simplify your answer to a single fraction.
b) Use differentiation from first principles to show that:
$$f ^ { \prime } ( x ) = - \frac { 1 } { ( 1 + x ) \sqrt { 1 - x ^ { 2 } } }$$
c) Sketch the curve on the axes provided over the page, showing clearly the behaviour of the curve near \(x = 0\) and \(x = \pm 1\). [0pt]
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[Question 2 - Continued] [0pt]
[Question 2 - Continued]
\includegraphics[max width=\textwidth, alt={}, center]{5023b2d9-ed3d-4a4a-b0c6-f529550b2e3e-09_1731_1566_913_260}
2. A function is defined by:
$$f ( x ) = \sqrt { \frac { 1 - x } { 1 + x } } , x \in \mathbb { R } , | x | < 1$$
a) P and Q are points on the curve with $x$-coordinates $x$ and $x + h$ respectively. Find the gradient of the line segment PQ . Simplify your answer to a single fraction.\\
b) Use differentiation from first principles to show that:
$$f ^ { \prime } ( x ) = - \frac { 1 } { ( 1 + x ) \sqrt { 1 - x ^ { 2 } } }$$
c) Sketch the curve on the axes provided over the page, showing clearly the behaviour of the curve near $x = 0$ and $x = \pm 1$.\\[0pt]
[Question 2 - Continued]\\[0pt]
[Question 2 - Continued]\\[0pt]
[Question 2 - Continued]\\
\includegraphics[max width=\textwidth, alt={}, center]{5023b2d9-ed3d-4a4a-b0c6-f529550b2e3e-09_1731_1566_913_260}\\
\hfill \mbox{\textit{SPS SPS FM 2022 Q2 [20]}}