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$Let f (x + y) = f (x) f (y) and f (x) = 1 + (\sin 2 x) g (x) where g (x) is continuous.$ $Then g^{'} (x) is equal to$

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f(x) g(0)

2g(0)

2f(x) g(0)

None of these

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detailed solution

Correct option is C

f(x)=limh→0fx+h−fxhlimh→0fxfh−fxh = fxlimh→0fh−1h =fxlimh→01+sin2hgh−1h =2fxlimh→0sin2h2hlimh→0gh = 2fx g0Hence (3)is the correct answer.

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Let f(x+y)=f(x)f(y) and f(x)=1+(sin⁡2x)g(x) where g(x) is continuous. Then g′(x) is equal to

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$Let f (x + y) = f (x) f (y) and f (x) = 1 + (\sin 2 x) g (x) where g (x) is continuous.$ $Then g^{'} (x) is equal to$