Math 250 3.4 Product and Quotient Rules, Population Growth Rates

Objectives

1) Find the derivative ofa product using the product rule !!__(i(x) · g(x)) = f'(x} · g(x) + j(x) · g'(x) dx

2) Simplify an expression (eventually resulting from a derivative) by factoring

a. GCF of each numerator

b. Least power (GCF) of each numerator

c. LCM of each denominator

3) Find the derivative of a product of three factors

!!_(J(x) · g(x) · h(x)) = f'(x) · g(x) · h(x) + f(x) · g'(x) · h(x) + j(x) · g(x) · h'(x) dx

4) Find the derivative of a quotient using the quotient rule !!__[! (x)J = g(x) · f'(x)- f(x) · g'(x) dx g(x) [g(x)]2

a. CAUTION: The order of the numerator is important - backward, and you have a sign error.

b. CAUTION: It is illegal to cancel g(x) from the first term, because there is no g(x) in the

second term also.

5) Where applicable, rewrite function before differentiating so the constant multiple rule applies but

not the quotient rule.

6) Use population growth models to

a. Calculate the instantaneous rate of change

b. Approximate when the rate of change is greatest.

c. Find the steady-state population

7) Finding derivatives using multiple rules

Examples and Practice:

1) Prove the product rule using the definition of the derivative. Explain why the derivative of the product is

not the product of the derivatives.

2) Simplifybyfactoring: 15 x3(x+t}X+ 15 x1 (x+t}% 4 2

3) Differentiate

f( ) xsinx a. x =--3

b. g(O) =5sin8cos8

4) Differentiate without multiplying flrst (use the 3-way product rule}: f (x) = (x' ~ 4x~-tJ1 <,

S) /(x) = x3

+5x+3 x2 -1

a. Find f'(x)

b. Find the equation of the tangent line to fat x = 2

6) Re-write each function so that the quotient rule will not be necessary when finding its derivative.

Sx2

- 3 b. y = ...!!. a. y= 4 3x5

7) The population of a culture of celts increases and approaches a constant level (called the steady state or

carrying capacity). The population is modeled by the function p(t) = 4nrf 1: +I), where t represents \..t +4

time measured in hours, and p is the number of cells at time t. The rate of change of the population is

called the growth rate.

a. Use GC to graph the population function. b. Find the derivative of the population function. c; Graph the derivative of the population function, and estimate the time when the growth rate is

greatest. d. Use the graph of the population function to estimate the steady-state population.

8) Differentiate.

(3x1 -2x-1Xx3 +5x) a. f(x)=

2

b.

c. f(x) = xcosx . x3 +2

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