Simple Rules for Differentiation
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Transcript of Simple Rules for Differentiation
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?