Question

Let $f\left(x\right)$ is a differentiable function on $x\in R$ , such that $f\left(x + y\right)=f\left(x\right)f\left(y\right)$ for all $x,y\in R$ where $f\left(0\right)\neq 0.$ If $f\left(5\right)=10,f^{'}\left(0\right)=6$ , then the value of $f^{'} \left(5\right)$ is equal to

Answer 1

$f^{'} \left(5\right) = \underset{h \rightarrow 0}{\text{lim}} \frac{f \left(5 + h\right) - f \left(5\right)}{h}$
$=\underset{h \rightarrow 0}{l i m}\frac{f \left(5 + h\right) - f \left(5 + 0\right)}{h}$
$=\underset{h \rightarrow 0}{l i m}\frac{f \left(5\right) . f \left(h\right) - f \left(5\right) . f \left(0\right)}{h}$
$\left[\right.\because f\left(x + y\right)=f\left(x\right).f\left(y\right)forallx,y\left]\right.$
$=\left(\underset{h \rightarrow 0}{l i m} \frac{f \left(h\right) - f \left(0\right)}{h}\right).f\left(5\right)$
$= f^{'} \left(0\right) . f \left(5\right) = 6 \times 10$
$=60$