# If $f(x+y)=f(x).f(y)$ for all x and y and if $f(5)=2$ and $f'(0) =3$ find $f'(5)$

$\begin{array}{1 1}(A)\;2 \\(B)\;6 \\(C)\;7 \\(D)\;none\;of\;these \end{array}$

we have , $f'(a)=\lim \limits_{h \to 0} \large\frac{f(a+h)-f(a)}{h}$
=> $f'(5) =\lim \limits_{h \to 0} \large\frac{f(5+h)-f(5)}{h}$
=> $\lim \limits_{h \to 0}\large\frac{f(5+h)-f(5+0)}{h}$
=> $\lim \limits_{h \to 0} \large\frac{f(5).f(h) -f(5).f(0)}{h}$
=> $f(5) \lim \limits_{h \to 0} \large\frac{f(0+h)-f(0)}{h}$
=> $f(5) .f'(0) =2.3$
$\qquad=6$
Hence B is the correct answer.