1. The composition function k(x) = (f o g)(x) = f (g(x)) g f g f R->[0,+oo)->[-1, 1]

2. Theorem 1 The chain rule g f g f R-->[0,+oo)-->[-1, 1] If k(x) =f (g(x)), then k’(x) = f ’ ( g(x) ) g’(x)

3. If k(x) =f (g(x)), then k’(x) = f ’ ( g(x) ) g’(x) k(x) = sin( x 2 ) k’(x) = cos ( x 2 ) 2x

4. If y=sin(3  t), find y’ A. 3  cos(3  t) B. cos(t) C. 3  t[cos(3  t)]

5. www.sendsteps.com Prepare to vote; keep your phone ready! TXT 1 2 Internet 1 2 Twitter 1 2 Voting is anonymous This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download-sendsteps-software.

6. If y=sin(3  t), find y’ A. 3  cos(3  t) B. cos(3  t) C. 3  t[cos(3  t)]

7. If y=sin(3  t), find y’ A. 3  cos(3  t) B. cos(3  ) C. 3  t[cos(3  t)] # votes: 0 Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software.

8. If y=sin(3pt), find y’ Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. A. B. C. 3p cos(3pt) cos(3p) 3pt[cos(3pt)] 33.3% 66.7% 100.0 % This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software. We will set these example results to zero once you've started your session and your slide show. In the meantime, feel free to change the looks of your results (e.g. the colors).

9. If y=sin(3  t), find y’ A. 3  cos(3  t) B. cos(3  t) C. 3  t[cos(3  t)]

10. Corallary k(x) = g n (x) = [g(x)] n k’(x) = n [g (x)] n-1 g’(x)

11. y=(2x+1) 4, find y’ A. 4(2)^3 B. 4(2x+1)^3 C. 8(2x+1) D. 8(2x+1)^3 # votes: 0 Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software.

12. y=(2x+1)4, find y’ Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. A. B. C. D. 4(2)^3 4(2x+1)^3 8(2x+1) 8(2x+1)^3 25.0% 50.0% 75.0% 100.0% This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software. We will set these example results to zero once you've started your session and your slide show. In the meantime, feel free to change the looks of your results (e.g. the colors).

13. If y=(2x+1) 4, find y’ A. 4(2) 3 B. 4(2x+1) 3 C. 8(2x+1) D. 8(2x+1) 3

14. If y=tan(sin(x)), find y’ A. -sec 2 [sin(x)]cos(x) B. sec 2 [sin(x)]cos(x) C. sec 2 [cos(x)] D. -csc 2 [sin(x)]cos(x)

15. If y=tan(sin(x)), find y’ A.1 B.2 C.3 D.4 # votes: 0 Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software.

16. If y=tan(sin(x)), find y’ A. B. C. D. 1 2 3 4 25.0% 50.0% 75.0% 100.0% Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software. We will set these example results to zero once you've started your session and your slide show. In the meantime, feel free to change the looks of your results (e.g. the colors).

17. If y=tan(sin(x)), find y’ A. -sec 2 [sin(x)]cos(x) B. sec 2 [sin(x)]cos(x) C. sec 2 [cos(x)] D. -csc 2 [sin(x)]cos(x)

18. If y=x cos(x 2 ), find dy/dx A. -x sin(x 2 ) + cos(x 2 ) B. -2x sin(x 2 ) + cos(x 2 ) C. -2x 2 sin(x 2 ) + cos(x 2 ) D. 2x 2 sin(x 2 ) + cos(x 2 )

19. If y=x cos(x 2 ), find dy/dx A.1 B.2 C.3 D.4 # votes: 0 Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software.

20. If y=x cos(x2), find dy/dx A. B. C. D. 1 2 3 4 25.0% 50.0% 75.0% 100.0% Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software. We will set these example results to zero once you've started your session and your slide show. In the meantime, feel free to change the looks of your results (e.g. the colors).

21. If y=x cos(x 2 ), find dy/dx A. -x sin(x 2 ) + cos(x 2 ) B. -2x sin(x 2 ) + cos(x 2 ) C. -2x 2 sin(x 2 ) + cos(x 2 ) D. 2x 2 sin(x 2 ) + cos(x 2 )

22. The chain rule If y = sin(u) and u(x) = x 2 then dy/dx = dy/du du/dx dy/du = cos(u) du/dx = 2x dy/dx = cos(u) 2x = cos(x 2 ) 2x

23. The chain rule If y = cos(u) and u(x) = x 2 + 3x then dy/dx = dy/du du/dx dy/du = -sin(u) du/dx = 2x + 3 dy/dx = -sin(u) (2x+3) = -sin(x 2 +2x) (2x+3)

24. y=tan(u) u = 10x – 5 find dy/dx A. -10 csc 2 (10x-5) B. sec 2 (10) C. -csc 2 (10x-5) D. 10 sec 2 (10x-5)

25. y=tan(u) u = 10x – 5 find dy/dx A.1 B.2 C.3 D.4 # votes: 0 Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software.

26. y=tan(u) u = 10x – 5 find dy/dx A. B. C. D. 1 2 3 4 25.0% 50.0% 75.0% 100.0% Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software. We will set these example results to zero once you've started your session and your slide show. In the meantime, feel free to change the looks of your results (e.g. the colors).

27. y=tan(u) u = 10x – 5 find dy/dx A. -10 csc 2 (10x-5) B. sec 2 (10) C. -csc 2 (10x-5) D. 10 sec 2 (10x-5)

28. Corallary k(x) = [3x 3 - x -2 ] 20 k’(x) = 20 [3x 3 - x -2 ] 19 (9x 2 +2x -3 )

29. If y = (sec(x)) 2 =sec 2 (x) find dy/dx A. 2 sec(x) tan(x) B. 2 sec 2 (x) tan(x) C. 2 sec(x) tan 2 (x) D. sec 2 (x) tan (x)

30. If y = (sec(x)) 2 =sec 2 (x) find dy/dx A.1 B.2 C.3 D.4 # votes: 0 Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software.

31. If y = (sec(x))2=sec2 (x) find dy/dx A. B. C. D. 1 2 3 4 25.0% 50.0% 75.0% 100.0% Closed InternetThis text box will be used to describe the different message sending methods. TXTThe applicable explanations will be inserted after you have started a session. TwitterIt is possible to move, resize and modify the appearance of this text box. This presentation has been loaded without the Sendsteps plug-in. Want to download the plug-in for free? Go to http://sendsteps.com/download- sendsteps-software. We will set these example results to zero once you've started your session and your slide show. In the meantime, feel free to change the looks of your results (e.g. the colors).

32. If y = (sec(x)) 2 =sec 2 (x) find dy/dx A. 2 sec(x) tan(x) B. 2 sec 2 (x) tan(x) C. 2 sec(x) tan 2 (x) D. sec 2 (x) tan (x)

33. Corallary =[3x 3 - x 2 ] 1/2 =[3x 3 - x 2 ] 1/2 k’(x) = ½ [3x 3 - x 2 ] -1/2 (9x 2 -2x)

34. Corallary k’(x) =

35. Corallary

36. If y = find dy/dx A.. B.. C..

37. If y = find dy/dx A.. B.. C..

38. Corallary = [sin(2x) ] 1/2 = [sin(2x) ] 1/2 k’(x) = ½ [sin(2x)] -1/2 (cos(2x) 2)

39. k(x) = sec(sin(2x)) k’(x) = sec(sin(2x))tan(sin(2x))(cos(2x) 2)

40. quizz 1.Write the equation of the line tangent to the graph of y = x – cos(x) when x=0. 2. Diff. g(x)=cot x[sin x – cos x]. 3. Find the x’s where the lines tangent to y= are horizontal.