Parameters of Individual Growth - Instantaneous Growth Rate		  
 
We now that animals, on a macroscopic scale, do not grow discrete, but continuously. Moreover, the growth rate changes during lifetime, usually it decreases with increasing size and age. Therefore, the growth rate computed by the methods introduced sofar is just an average for the time between t1 and t2 (Fig. A).
 
To make our measure of the growth rate more accurate, we can reduce the distance between t1 and t2. What happens, however, if the distance in time between t1 and t2 becomes zero? Well, this is simple differentiation business. The gradient of any curve at any point is equal to the gradient of the tangent at this point (Fig. B), and it is computed by the 1st derivative of the corresponding function.
 

Rate of growth at point t
 = 1st derivative of growth function
  = dM/dt

The same holds true for the relative growth rate, which is called "instantaneous" now:

Instantaneous relative rate of growth GM.t = (1/Mt) * dM/dt

If we do not know the growth function, integration of this equation leads to

dM/Mt = GM dt <=> ln(Mt) = GM * t + c

and, after some formula shuffling, G may be approximated by

GM = ln(M2/M1) / (t2 - t1)

Regarding our crab example, G = ln(10.0/6.0)/(30-1) = 0.018.

If we do know the growth function Mt = fM(t), things are much easier, because

GM.t = (1/Mt) * 1st derivative of growth function fM(t)
 
 
If the relation between size S and body mass M is exponential,
M = a * Sb <=> ln(M) = ln(a) + b * ln(S)

then the mass specific instantaneous rate GM can be derived easily from the size specific rate GS by

GM = b * GS
 
The instantaneous rate of growth plays an important role in production studies.
Derivatives of commonly used growth functions are given here.
  See graphical presentation of growth rates.