We are given $R(t)$ = $P(X>t)$ for all $x > 0$ and
$R(0) = 1 - Fx(0) = 1\text{ and }\lim\limits_{t \to \infty} R(t) = 0$
The random variable $X$ also satisfies the memoryless property:
$P(X>s+t|X>t) = P(X>s)\text{ for }s>0\text{ and }t>0$
Let R'(0) = - \lambda\ where \ \lambda\ , is some positive constant. I need to show that X must be exponentially distributed.
Given that $\dfrac{R(t + h) - R(t)}{h}$ = $R(t)\left[\dfrac{R(h) - 1}{h}\right]$
Show that by letting $\lim\limits_{h\to \infty}$ $\dfrac{dR(t)}{dt} = -\lambdaR(t)$ (I think we should use Hopital's rule here I am not sure by differentiating $\left[\dfrac{R(h) - 1}{h}\right]$ and letting $h$ tend to $0$, we will get $-\lambda$ for this part but I got stuck afterwards).
Also argue that $X$ is an exponential random variable with rate parameter $\lambda$ by solving the differential equation above respecting the conditions:
R(0) = 1 - Fx(0) = 1\text{ and }\lim\limits_{t \to \infty} R(t) = 0$$