Let f : R → R be a function given by $f(x+y)=f(x) f(y)$  for all $x, y \in R$. If $f(x)=1+x g(x)+x^2 g(x) \phi(x)$ such that $\lim\limits_{x \rightarrow 0} g(x)=a$ and $\lim\limits_{x \rightarrow 0} \phi(x)=b$, then f'(x) is equal to

a f(x)

We have,

$f'(x) =\lim\limits_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$

$\Rightarrow f'(x) =\lim\limits_{h \rightarrow 0} \frac{f(x) f(h)-f(x)}{h}$        [∵ f(x + y) = f(x) f(y)]

$\Rightarrow f'(x) =\lim\limits_{h \rightarrow 0} f(x)\left(\frac{f(h)-1}{h}\right) $

$\Rightarrow f'(x) =f(x) \lim\limits_{h \rightarrow 0}\left\{\frac{1+h g(h)+h^2 g(h) \phi(h)-1}{h}\right\}$

$\Rightarrow f'(x) =f(x) \lim\limits_{h \rightarrow 0}\{g(h)+h g(h) \phi(h)\}=a f(x)$