N(t)=130,000(0.29)^t Find N'(t)

Question

In one city, 29% of all aluminum cans distributed will be recycled each year. A juice company distributes 130,000 cans. The number still in use after time t, in years, is given by N(t)=130,000(0.29)^t. Find N'(t).

Answer 1

Question: In one city, 29% of all aluminum cans distributed will be recycled each year. A juice company distributes 130,000 cans. The number still in use after time t, in years, is given by N(t)=130,000(0.29)^t. Find N'(t).

What you have is a function of t, N(t), and you want to differentiate it to get N'(t).

The problemis: you have a function like y = a^x, so how do you differentiate that?

Solution

Use logs.

let y = a^x   -- now take logs of both sides

ln(y) = x.ln(a)   -- now differntiate both sides, wrt x

(1/y).y' = ln(a)

y' = y.ln(a)

y' = a^x.ln(a)

Since we have N(t)=130,000(0.29)^t, then

N'(t) = 130,000(0.29)^t.ln(0.29)