# Chapter 3 - Section 3.4 - The Chain Rule - 3.4 Exercises: 86

*The rate of increase in world population in 1920 is $26.255mi/year$. *The rate of increase in world population in 1950 is $39.783 mi/year$. *The rate of increase in world population in 2000 is $79.53mi/year$.

#### Work Step by Step

$$P(t)=(1436.53)(1.01395)^t$$ 1) Here we again encounter the rate of increase, which means rate of change, in the question. Therefore, we need to find the derivative of $P(t)$. $$P'(t)=1436.53[(1.01395)^t]'$$ $$P'(t)=1436.53\ln(1.01395)(1.01395)^t$$ $[(a^t)'=\ln a\times a^t]$ $$P'(t)=19.901(1.01395)^t(mi/year)$$ 2) $t=0$ corresponds to the year 1900, then 1920 corresponds to $t=20$, 1950 to $t=50$ and 2000 to $t=100$. *The rate of increase in world population in 1920 is $$P'(20)=19.901(1.01395)^{20}\approx26.255(mi/year)$$ *The rate of increase in world population in 1950 is $$P'(50)=19.901(1.01395)^{50}\approx39.783(mi/year)$$ *The rate of increase in world population in 2000 is $$P'(100)=19.901(1.01395)^{100}\approx79.53(mi/year)$$

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