Question

Let $f(x y)=f(x) f(y)$ for all $x, y \in R$. If $f^{\prime}(1)=2$ and $f(4)=4$ then $f^{\prime}(4)$ equal to

Answer 1

$f ( x )= x ^{ k }$ satisfies given relation
$f (4)=4 \Rightarrow 4^{ k }=4^{\prime}$
$k =1$
Now, $f ^{\prime}(1)=2 $
$\Rightarrow \displaystyle\lim _{ h \rightarrow 0} \frac{ f (1+ h )- f (1)}{ h }=2$
$\Rightarrow \displaystyle\lim _{ h \rightarrow 0} \frac{ f (1) f ( h )- f (1)}{ h }=2$
$\Rightarrow f (1) \displaystyle\lim _{ h \rightarrow 0} \frac{ f ( h )- f (1)}{ h }=2$
$\Rightarrow \displaystyle\lim _{ h \rightarrow 0} \frac{ f ( h )- f (1)}{ h }=2($ using $f (1)=1)$
Now, $f^{\prime}(4)=\displaystyle\lim _{h \rightarrow 0} \frac{f(4+h)-f(4)}{h}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{f(4) f(h)-f(4)}{h}$
$=\left(\displaystyle\lim _{h \rightarrow 0} \frac{f(h)-1}{h}\right) f(4)=2 f(4) ($ using $f(1)=1)$
$=2 \times 4=8$