Implicit Differentiation Objective: To find derivatives of functions that we cannot solve for y.
Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find...
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Transcript of Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find...
![Page 1: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/1.jpg)
Simple Rules for Differentiation
![Page 2: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/2.jpg)
Objectives
Students will be able to• Apply the power rule to find
derivatives.• Calculate the derivatives of sums
and differences.
![Page 3: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/3.jpg)
Rules
Power Rule• For the function ,
for all arbitrary constants a.
axxf )( 1)( aaxxf
![Page 4: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/4.jpg)
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
![Page 5: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/5.jpg)
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
![Page 6: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/6.jpg)
Example 1
Use the simple rules of derivatives to find the derivative of
6)( xxf
![Page 7: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/7.jpg)
Example 2
Use the simple rules of derivatives to find the derivative of
23
10)( ppD
![Page 8: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/8.jpg)
Example 3
Use the simple rules of derivatives to find the derivative of
4
6x
y
![Page 9: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/9.jpg)
Example 4
Use the simple rules of derivatives to find the derivative of
23 156 xxy
![Page 10: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/10.jpg)
Example 5
Use the simple rules of derivatives to find the derivative of
ttttp
5612)( 4
![Page 11: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/11.jpg)
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
![Page 12: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/12.jpg)
Example 7
Find all value(s) of x where the tangent line to the function below is horizontal.
365)( 23 xxxxf
![Page 13: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/13.jpg)
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
![Page 14: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/14.jpg)
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.
![Page 15: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/15.jpg)
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.
![Page 16: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/16.jpg)
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.
![Page 17: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/17.jpg)
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?
![Page 18: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/18.jpg)
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.
![Page 19: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/19.jpg)
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
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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
![Page 21: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/21.jpg)
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
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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
![Page 23: Simple Rules for Differentiation. Objectives Students will be able to Apply the power rule to find derivatives. Calculate the derivatives of sums and.](https://reader036.fdocuments.us/reader036/viewer/2022062422/56649e905503460f94b95739/html5/thumbnails/23.jpg)
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?