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Let f(x) be a function satisfying  \mathrm{f(x+y)=f(x) f(y)}   for all \mathrm{x, y \in R~and ~f(x)=1+x g(x)}  where  \mathrm{ \lim _{x \rightarrow 0} g(x)=1} . Then\mathrm{ f^{\prime}(x)} is equal to

Option: 1

\mathrm{g^{\prime}(x)}


Option: 2

\mathrm{g(x)}


Option: 3

\mathrm{f(x)}


Option: 4

none of these


Answers (1)

best_answer

\mathrm{ f^{\prime}(x) =\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim _{h \rightarrow 0} \frac{f(x) f(h)-f(x)}{h} }
\mathrm{ =f(x) \lim _{h \rightarrow 0} \frac{f(h)-1}{h} \quad[\because f(x+y)=f(x) f(y)] }

\mathrm{ =f(x)\left(\lim _{h \rightarrow 0} \frac{1+h g(h)-1}{h}\right) \quad\left[\begin{array}{l} \text { Using: } f(x) \\ =1+x g(x) \end{array}\right] \\ }
\mathrm{ =f(x) \lim _{h \rightarrow 0} g(h)=f(x): 1=f(x) . }

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