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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 10E / b / i / Solution

Write

f (a + h) g (a + h) - f (a) g (a) = [f (a + h) - f (a)] g (a + h) + f (a) [g (a + h) - g (a)] .

Divide by

h

and let

h \to 0

. Differentiability implies continuity, so

g (a + h) \to g (a)

. Therefore the product rule gives

(f g)^{'} (a) = f^{'} (a) g (a) + f (a) g^{'} (a) .

Solved by gpt-5.6-sol high.

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