#### To determine

**To estimate:** The total personal income was rising in the year 1999.

#### Answer

The total personal income was rising by about $1,627,415,600 per year in 1999.

#### Explanation

**Derivative rule:**

Product Rule: ddx[f1(x)f2(x)]=f1(x)ddx[f2(x)]+f2(x)ddx[f1(x)]

**Calculation:**

Let *x* be the number of year, f(x) be the function of growth of population and g(x) be the function of growth of average income.

f(x)=(Population increased per year)×(Number of years)+(Population in 1999)=(9200)×(x)+(961,400)=9200x+961,400

g(x)=(National average per year)×(Number of years)+(Average annual income per capita)=(1225)×(x)+(30,593)=1225x+30,593

Since the population was increasing at roughly 9200 per year and the average annual income was increasing at about $1400 per year, f′(x)=9200 and g′(x)=1400.

In 1999, f(0)=961,400, g(0)=30,593, f′(0)=9200 and g′(0)=1400.

Obtain the total personal income was rising in the year 1999.

Let F(x) be total personal income which is product of f(x) and g(x).

That is, F(x)=f(x)g(x).

Apply the product rule and simplify the terms,

F′(x)=ddx(f(x)g(x))=f(x)ddx(g(x))+g(x)ddx(f(x))=f(x)g′(x)+g(x)f′(x)

Substitute 0 for *x,*

F′(0)=f(0)g′(0)+g(0)f′(0)

Substitute the values f(0)=961,400, g(0)=30,593, f′(0)=9200 and g′(0)=1400,

F′(0)=(9200(0)+961,400)1400+(1225(0)+30,593)9200=961,400×1400+30,593×9200=1,346,960,000+281,455,600=1,627,415,600

Therefore, the total personal income was rising by about $1,627,415,600 per year in 1999.

**To Explain:** The meaning of each term in the product.

**Explanation:**

In 1999, the term f(0)g′(0)=$1,346,960,000 represents the portion of the rate of change of total income due to existing population’s increasing income.

In 1999, the term f′(0)g(0)=$281,455,600 represents the portion of the rate of change of total income due to increasing population.