Population growth Consider the following population functions.
b. What is the instantaneous growth rate at t=5?
p(t) = 600 (t²+3/t²+9)

Verified step by step guidance

1. Step 1: Understand that the instantaneous growth rate of a population function p(t) at a specific time t is given by the derivative of the function, p'(t), evaluated at that time.
2. Step 2: Identify the given population function: p(t) = 600 \left(\frac{t^2 + 3}{t^2 + 9}\right).
3. Step 3: Use the quotient rule to differentiate p(t). The quotient rule states that if you have a function in the form of \(\frac{u(t)}{v(t)}\), its derivative is \(\frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2}\). Here, u(t) = t^2 + 3 and v(t) = t^2 + 9.
4. Step 4: Compute the derivatives u'(t) and v'(t). For u(t) = t^2 + 3, u'(t) = 2t. For v(t) = t^2 + 9, v'(t) = 2t.
5. Step 5: Substitute u(t), v(t), u'(t), and v'(t) into the quotient rule formula to find p'(t). Then, evaluate p'(t) at t = 5 to find the instantaneous growth rate.