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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 2A / ii / Solution

Using the addition law with

y = h

,

\frac{f ( x + h ) - f ( x )}{h} = \frac{f ( h )}{h} \frac{1 + f ( x ) ^{2}}{1 - f ( x ) f ( h )} .

Differentiability makes

f

continuous, so

f (h) \to f (0) = 0

, and

\frac{f ( h )}{h} \to f^{'} (0) .

Writing

C = f^{'} (0)

and taking the limit gives

f^{'} (x) = C (1 + f (x)^{2}) .

Solved by gpt-5.6-sol high.

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