Let fx-y-fxfy and fx-1-xgxGx- where limx- 0gx-a and limx- 0Gx-b- Then f-x is equal to

The correct option is D abf(x) f'(x)=Lim_h→0f(x+h) f(x)h =Lim_h→0 f(x)(f(h) 1h) #160; f'(x)=f(x) #160; f'(0) =f(x) #160; #160; ...

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Byju's Answer

Standard XII

Mathematics

Algebra of Derivatives

Let fx+y=fx...

Question

Let $f (x + y) = f (x) f (y)$ and $f (x) = 1 + x g (x) G (x)$ , where
$lim x \to 0 g (x) = a$ and $lim x \to 0 G (x) = b$ . Then $f^{'} (x)$ is equal to

1 + a b

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a b

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f (x)

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a b f (x)

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Solution

The correct option is D $a b f (x)$
$f^{'} (x) = L i m_{h \to 0} \frac{f (x + h) - f (x)}{h}$
$= L i m_{h \to 0} f (x) (\frac{f (h) - 1}{h})$
$f^{'} (x) = f (x) f^{'} (0)$
$= f (x) f^{'} (0)$
$f^{'} (x) = g (x) G (x) + x g^{'} (x) G (x) + x g (x) G^{'} (x)$
$f^{'} (0) = a b + 0 + 0$
$f^{'} (x) = (a b) f (x)$
Hence,option 'D' is correct.

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Similar questions

Q. Let

f (x + y) = f (x) f (y) a n d f (x) = 1 + x g (x) G (x)

, where

l i m x \to 0 g (x) = a

and

l i m x \to 0 G (x) = b

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Q. Let

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be a function satisfying

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for all

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and

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

, where

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for all

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Let f(x+y)=f(x)f(y) and f(x)=1+xg(x)G(x), wherelimx→0g(x)=a and limx→0G(x)=b. Then f′(x) is equal to

The correct option is D abf(x)f′(x)=Limh→0f(x+h)−f(x)h=Limh→0 f(x)(f(h)−1h) f′(x)=f(x)f′(0)=f(x)f′(0)f′(x)=g(x)G(x)+xg′(x)G(x)+xg(x)G′(x)f′(0)=ab+0+0f′(x)=(ab)f(x)Hence,option 'D' is correct.

Let $f (x + y) = f (x) f (y)$ and $f (x) = 1 + x g (x) G (x)$ , where
$lim x \to 0 g (x) = a$ and $lim x \to 0 G (x) = b$ . Then $f^{'} (x)$ is equal to