What is the formula of the exponential population growth curve?

The correct answer is Option (2) → dN/dt = (b – d) × N

When resources are unlimited, a population grows exponentially. If b is the per capita birth rate and d is the per capita death rate, the change in population size N per unit time is the difference between them multiplied by N.

Option (1) dN/dt = N — Incorrect. This leaves out the birth and death rates. Population growth depends on how many are born and how many die per individual, so N alone cannot give the rate of change.

Option (2) dN/dt = (b – d) × N — Correct. The increase or decrease in N per unit time is (b – d) × N. Writing (b – d) as r, the same equation becomes dN/dt = rN.

Option (3) dN/dt = (b – d) × r — Incorrect. (b – d) is itself r, so (b – d) × r would multiply r by r. The growth rate must be multiplied by the population size N.

Option (4) dN/dt = (b + d) × N — Incorrect. Deaths reduce the population, so the death rate is subtracted from the birth rate, not added to it.

NCERT Extract:
Growth Models – “… Then the population grows in an exponential or geometric fashion. If in a population of size N, the birth rates (not total number but per capita births) are represented as b and death rates (again, per capita death rates) as d, then the increase or decrease in N during a unit time period t (dN/dt) will be dN/dt = (b – d) × N Let (b–d) = r, then dN/dt = rN”