Find the rate of change

Question

given p(t) = 48/(2+10e^-t)

1. Find the rate of change at time t
2. evaluate lim t->infinity p'(t)

Answer 1

Use the power rule and the chain rule.

p(t) = 48 / (2 + 10e^-t) = 48 (2 + 10e^-t)^-1

p'(t) = 48(-1)(2 + 10e^-t)^-2 (-10e^-t)

p'(t) = 480e^-t / (2 + 10e^-t)²

As t approaches ∞, the numerator approaches zero and the denominator approaches 4, so p'(t) approaches zero as t approaches ∞.