instantaneous rate of growth

Question

Populations of two species $A$ and $B$ at time $0$ are equal. If the instantaneous rates of growth of populations of species $A$ and $B$ are $u$ and $u + 1$ respectively, $u > 0$, then at time $1$ the population of species $B$ would be

(a) twice the population of species $A$

(b) $log 10$ times of the population of species $A$

(c) $e^{u}$ times the population of species $A$

(d) $e$ times the population of species $A$

I tries to understand instantaneous growth on the inernet but i found it very confusing, any help will be appreciated!

Answer 1

Hint:

Population at time t : $N_o e^{\lambda t}$ where $N_o$ is the initial population and $\lambda$ is the rate of growth.

Thus the answer should be $e$ times the population of $A$

This was derived by:

$$\frac{dP}{dt}=k P$$ $$\int\frac{dP}{P}=\int k dt$$ $$\ln P=k t$$

Note, that population growth is proportional to current population