Since given increasing,so $N'(t)>0$ but what will be $N''(t)$ for the slow rate part?
Second derivative of a function tells about the concavity of the graph of f and the rate of change of the graph of f'.
Go to the above link you will find that if f''(x) < 0 then the graph of f is concave down and if f''(x) > 0 then graph of f is concave up.
I am not getting what you want to say by the slope will be a bit less here the slope of the graph decreases as the value of x(time) increases.
thnx @ Kushagra Chatterjee