Question

A function $f:R⟶R$ satisfies the equation $f(x+y)=f\left(x\right),f\left(y\right)$ for all $x,yϵR,f\left(x\right)\ne 0$ Suppose that the function is differentiable at x=0 and $f^{{}^{\prime }}\left(0\right)=2$ prove that $f^{{}^{\prime }}\left(x\right)=2f\left(x\right)$

Answer 1

$f^{\prime }\left(x\right)={lim}_{x\to 0}\frac{f(x+b)-f\left(x\right)}{h}$

$={lim}_{x\to 0}\frac{f\left(x\right)f\left(h\right)-f\left(x\right)}{h}\Rightarrow {lim}_{x\to 0}f\left(x\right)\left[\frac{f\left(x\right)-1}{h}\right]$

$={lim}_{x\to 0}f\left(x\right)\left[\frac{f\left(h\right)-f\left(0\right)}{h}\right]\because f(x+0)=f\left(x\right)\times f\left(0\right)\Rightarrow f\left(0\right)=1$

$f^{\prime }\left(x\right)=f\left(x\right)f^{\prime }\left(0\right)$

$f^{\prime }\left(x\right)=2f\left(x\right)$ $\because f^{\prime }\left(x\right)=2$