Question

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

Answer 1

Answer: 8

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