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  • Let be a function satisfying <img alt="\mathrm{f(x+y)=f(x) f(y)}" src="https://entrancecorn

Let \mathrm{f(x)} be a function satisfying \mathrm{f(x+y)=f(x) f(y)} for all \mathrm{x, y \in \mathbf{R}} and \mathrm{f(x)=1+x g(x)} where \mathrm{\lim _{x \rightarrow 0} g(x)=1\: then\: 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^{}(x) }
\mathrm{=\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}=f(x) \lim _{h \rightarrow 0} \frac{1+h g(h)-1}{h} }

\mathrm{=f(x) \lim _{h \rightarrow 0} g(h)=f(x) }

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