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)