At time t (in days), the size S of a population of butterflies is given by the formula s= 600/1+49(0.6)^t.

Question

When does the maximum growth rate occur?

Answer 1

I will assume you meant:

s = 600/(1 + 49(.6)^t)
= 600(1 + 49(.6)^t)^-1

then growth rate = s'
= -600(1+49(.6)^t)^-2 (49ln.6)(.6)^t

now you want the max of growth rate, so
you need s'' and set that equal to zero and solve

I suggest taking the product rule, and carefully doing the algebra. It's going to be messy!