Question

Let $f$ be a differentiable function such that $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x,y\in R$. If $f\left(0\right)=1,f\left(3\right)=3$ and $f^{\prime }\left(0\right)=11$. Then the value of $f^{\prime }\left(3\right)$ is equal to

Answer 1

We have,
$f^{\prime }\left(x\right)=\underset{h\to 0}{lim}\frac{f(x+h)-f\left(x\right)}{h}$
$\Rightarrow f^{\prime }\left(3\right)=\underset{h\to 0}{lim}\frac{f(3+h)-f\left(3\right)}{h}$
$=\underset{h\to 0}{lim}\frac{f\left(3\right)f\left(h\right)-3}{h}$
$=3\times \underset{h\to 0}{lim}\left(\frac{f\left(h\right)-1}{h}\right)$
$=3\times \underset{h\to 0}{lim}\left(\frac{f\left(h\right)-f\left(0\right)}{h}\right)$
$=3f^{\prime }\left(0\right)$
$=3\times 11=33$