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Theorems for Differentiability

Let f x + y =...

Question

Let f(x + y) = f(x)f(y) and f(x) = 1 + sin(3x)g(x) where g(x) is continuous then f'(x) is
[Kerala (Engg.) 2005]

A

f(x)g(0)

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B

3g(0)

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C

f(x)cos 3x

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D

3 f(x) g(0)

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E

3 f(x)g(x)

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Solution

The correct option is C f(x)cos 3x
f(x) = 1 + sin(3x)g(x)
f'(x) = 3 cos 3x g(x) + sin 3 x g' (x) = f(x) cos 3x.

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Similar questions

Q. Let f(x + y) = f(x)f(y) and f(x) = 1 + sin(3x)g(x) where g(x) is continuous then f'(x) is
[Kerala (Engg.) 2005]

f (x + y) = f (x) f (y)

f (x) = 1 + (sin 2 x) g (x)

g (x)

is continuous, then

f^{'} (x)

f (x + y) = f (x) + f (y)

f (x) = x^{2} g (x)

x, y ϵ R

g (x)

is continuous function. Then

f^{'} (x)

f (x)

be continuous and differentiable function satisfying

f (x + y) = f (x) f (y)

x, y ϵ R

f (x)

can be expressed as

f (x) = 1 + x p (x) + x^{2} q (x)

lim x \to 0 p (x) = a

lim x \to 0 q (x) = b

f^{'} (x)

f (x + y) = f (x) f (y)

f (x) = 1 + x g (x) G (x)

lim x \to 0 g (x) = a

lim x \to 0 G (x) = b

f^{'} (x)

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