Answer
The rate at which total personal income was rising in 1999 was $~~\$1,627,415,600/year$
The expression for the product rule is:
$\frac{d~TPI}{dt} = (9200)(30,593+1400~t)+(961,400+9200~t)(1400)$
$9200$ is the rate of change of the population
$(961,400+9200~t)$ is the population $t$ years after 1999
$1400$ is the rate of change of the average income
$(30,593+1400~t)$ is the average income $t$ years after 1999
Work Step by Step
Let $P$ be the population $t$ years after 1999:
$P = 961,400+9200~t$
Let $I$ be the average income $t$ years after 1999:
$I = 30,593+1400~t$
We can find an expression for the total personal income $t$ years after 1999:
$TPI = (961,400+9200~t)(30,593+1400~t)$
We can use the product rule to find the rate at which total personal income was rising:
$\frac{d~TPI}{dt} = (9200)(30,593+1400~t)+(961,400+9200~t)(1400)$
$\frac{d~TPI}{dt} = 25,760,000~t+1,627,415,600$
Note that $t=0$ in the year 1999.
The rate at which total personal income was rising in 1999 was $~~\$1,627,415,600/year$
The expression for the product rule is:
$\frac{d~TPI}{dt} = (9200)(30,593+1400~t)+(961,400+9200~t)(1400)$
$9200$ is the rate of change of the population
$(961,400+9200~t)$ is the population $t$ years after 1999
$1400$ is the rate of change of the average income
$(30,593+1400~t)$ is the average income $t$ years after 1999
