Simple Rules for Differentiation

Objectives

Students will be able to• Apply the power rule to find

derivatives.• Calculate the derivatives of sums

and differences.

Rules

Power Rule• For the function ,

for all arbitrary constants a.

axxf )( 1)( aaxxf

Rules

Sums and Differences Rule• If both f and g are differentiable at x, then

the sum and the difference are differentiable at x and the derivatives are as follows.

gf gf

)()()(

derivativeahas

)()()(

xgxfxF

xgxfxF

Rules

Sums and Differences Rule• If both f and g are differentiable at x, then

the sum and the difference are differentiable at x and the derivatives are as follows.

gf gf

)()()(

derivativeahas

)()()(

xgxfxG

xgxfxG

Example 1

Use the simple rules of derivatives to find the derivative of

6)( xxf

Example 2

Use the simple rules of derivatives to find the derivative of

23

10)( ppD

Example 3

Use the simple rules of derivatives to find the derivative of

4

6x

y

Example 4

Use the simple rules of derivatives to find the derivative of

23 156 xxy

Example 5

Use the simple rules of derivatives to find the derivative of

ttttp

5612)( 4

Example 6

Find the slope of the tangent line to the graph of the function at x = 9. Then find the equation of the tangent line.

25 34 xxy

Example 7

Find all value(s) of x where the tangent line to the function below is horizontal.

365)( 23 xxxxf

Example 8

Assume that a demand equation is given by

Find the marginal revenue for the following levels (values of q). (Hint: Solve the demand equation for p and use the revenue equation R(q) = qp .)

pq 1005000

a. q = 1000 units

b. q = 2500 units

c. q = 3000 units

Example 9-1

An analyst has found that a company’s costs and revenues in dollars for one product are given by the functions

and

respectively, where x is the number of items produced.

xxC 2)(

10006)(

2xxxR

a. Find the marginal cost function.

Example 9-2

An analyst has found that a company’s costs and revenues in dollars for one product are given by the functions

and

respectively, where x is the number of items produced.

xxC 2)(

10006)(

2xxxR

b. Find the marginal revenue function.

Example 9-3

An analyst has found that a company’s costs and revenues in dollars for one product are given by the functions

and

respectively, where x is the number of items produced.

xxC 2)(

10006)(

2xxxR

c. Using the fact that profit is the difference between revenue and costs, find the marginal profit function.

Example 9-4

An analyst has found that a company’s costs and revenues in dollars for one product are given by the functions

and

respectively, where x is the number of items produced.

xxC 2)(

10006)(

2xxxR

d. What value of x makes the marginal profit equal 0?

Example 9-5

An analyst has found that a company’s costs and revenues in dollars for one product are given by the functions

and

respectively, where x is the number of items produced.

xxC 2)(

10006)(

2xxxR

e. Find the profit when the marginal profit is 0.

Example 10-1

The total amount of money in circulation for the years 1915-2002 can be closely approximated by

where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.

1394335.142746.379044.3)( 23 ttttM

a. 1920

Example 10-2

The total amount of money in circulation for the years 1915-2002 can be closely approximated by

where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.

1394335.142746.379044.3)( 23 ttttM

b. 1960

Example 10-3

The total amount of money in circulation for the years 1915-2002 can be closely approximated by

where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.

1394335.142746.379044.3)( 23 ttttM

c. 1980

Example 10-4

The total amount of money in circulation for the years 1915-2002 can be closely approximated by

where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.

1394335.142746.379044.3)( 23 ttttM

d. 2000

Example 10-5

The total amount of money in circulation for the years 1915-2002 can be closely approximated by

where t represents the number of years since 1900 and M(t) is in millions of dollars. Find the derivative of M(t) and use it to find the rate of change of money in circulation in the following years.

1394335.142746.379044.3)( 23 ttttM

e. What do your answers to parts a-d tell you about the amount of money in circulation in those years?