1. By: Hunter Faulk Anthony Stephens Meghan Troutman

2. Defines the rate of change in a given function. Lim f(x+h) – f(x) h→0 h Example: f(x)= f’(x)=2x http://images.google.com/

3. Newton and Leibniz shared credit for the development of the differential and integral calculus. They both contributed in the math world by developing new areas of calculus such as differentiation, integration, curves, and optics. Both published their facts that included integral and derivative notation we still use today.

4. The Derivative of the Functions Will Use Notation That Depends on the Function FunctionFirst DerivativeSecond Derivative F(x)F’(x)F’’(x) G(x)G’(x)G’’(x) YY’ or dy dx Y’’ or

5. Definition of the Power Rule The Power Rule of Derivatives gives the following: For any real number n,the derivative of f(x) = x n is f ’(x) = nx n-1 which can also be written as Definition of the quotient rule Rule for finding the derivative of a quotient of two functions. If both f and g are differentiable, then so is the quotient f(x)/g(x). In abbreviated notation, it says (f/g)′ = (gf′ − fg′)/g 2.

6. Definition of the product rule Rule for finding the derivative of a product of two functions. If both f and g are differentiable, then (fg)′ = fg′ + f′g. Chain rule The formula is. Another form of the chain rule is.

7. Rule number 1 If y=, then Example If y=, then www.open.salon.com

8. Number one y= Number two y=

9. Work Page

10. Answers to Power Rule Problems Problem One Problem Two Original Function is blue. Derivative is red.

11. Used when given composite functions. A composite function is a function inside another function. F(G(x)) First you take the derivative of the outside function, while leaving the inside function alone. Then you multiply this by the derivative of the inside function, with respect to its variable x. If y= f(g(x)), then y’=

12. Chain Rule Examples Problem one Problem two www.babble.com

13. Work Page

14. Answer to Chain Rule Problem One www. school-clipart.com Problem one

15. Answer to Chain Rule Problem Two Problem two

16. Bottom function times the derivative of the top minus the top function times the derivative of the bottom. Then divide the whole thing by the bottom function squared. If f(x)=, then f’(x)=

17. Quotient Rule Examples Problem one Problem two

18. Work Page

19. Problem one

20. Answer to Quotient Problem Number Two Problem Two (x -2 + x -6 )(-3x -4 + 8x -9 )– (x -3 - x -8 )(-2x -3 - 6x -7 ) (x -2 + x -6 ) 2

21. When a function involves two terms multiplied together, we use the Product Rule. To find the derivative of two things multiplied by each other, you multiply the first function by the derivative of the second, and add that to the second function multiplied by the derivative of the first. If F(x) = uv, then f’(x) = u

22. Product Rule Examples www.first90days.wordpress.com Problem One f(x)= (9x 2 +4x)(x 3 -5x 2 ) Problem Two (x 5 -11x 8 )

23. Work Page

24. F’(x)= (9x 2 +4x)(3x 2 -10x) + (x 3 -5x 2 )(18x+4)

25. Answer to Product Rule Problem Two Y’= (5x 4 -88x 7 ) + (x 5 – 11x 8 ) www.schools.sd68.bc.ca

26. Implicit differentiation is the procedure of differentiating an implicit equation with respect to the desired variable while treating the other variables as unspecified functions of x. taking d of another variable Y2 -> 2y

27. Examples of Implicit Differentiation Problem One X 2 + y 2 = 5 Problem Two X 2 + 3xy + y 2 = ∏

28. Work Page

29. Solving Implicit Differentiation Problem One 2x =2y = 0 2x = -2y = www. school-clipart.com

30. Answer to Implicit Differentiation Problem Two 2x + 3x + 3y + 2y = 0 (3x +2y) = -2x - 3y = -2x – 3y 3X +2Y

31. A method for finding the derivative of functions such as y = xsin x and Y= lny=xln3 =x(0)+ ln3(1) =ln3 = ln3

32. Examples of Logarithmic Differentiation Problem one y= Problem two y= http://images.google.com/

33. Work Page

34. Y= lny=xlnx =x( )+lnx = (1+lnx)

35. Answers to Logarithmic Differentiation Problem Two Y= =sinx + lnxcosx = ( +lnx(cosx)) www.schools.sd68.bc.ca

36. 1978 AB 2 Let f(x) = (1 - x)2 for all real numbers x, and let g(x) = ln(x) for all x > 0. Let h(x) = ( 1 - ln(x))2. a. Determine whether h(x) is the composition f(g(x)) or the composition g(f(x)). b. Find h′ (x). c. Find h″(x). d. On the axes provided, sketch the graph of h.

37. FRQ Answer A)f(g(x)) = f(lnx) = (1-lnx)2 g(f(x)) = g((1-x)2) = ln((1-x)2) Therefore h(x) = f(g(x)) B) h’(x) = 2(1-lnx)(1/x) = 2 lnx-1 C) h’’(x) = 2 x(1/x) – (lnx-1) x2 = 2 * 2 – lnx x2 D) The Inflection point is at (-e2, 1) The minimum is at (e, 0 )

38. FRQ Example 1977 AB 7 BC 6 Let f be the real-valued function defined by f(x) = sin3(x) + sin3|x|. a. Find f′(x) for x > 0. b. Find f’(x) for x < 0. c. Determine whether f(x) is continuous at x = 0. Justify your answer. d. Determine whether the derivative of f(x) exists at x = 0. Justify your answer.

39. FRQ Answer A) For x > 0 F(x) = sin 3 x + sin 3 x = 2sin 3 x F’(x) = 6sin 2 xcosx B) For x > 0 f(x) = sin 3 x + sin 3 (-x) = sin 3 x - sin 3 x = 0 F’(x) = 0 C) f(0) = 0 Lim f(x) = lim 2sin 3 x = 0 Lim f(x) = lim 0 = 0 Since Lim f(x) = 0 = f(0), the function of f is continous at x = 0 D) F’(x) = lim f(x+h) – f(x) if the limit exists h At x = 0 Lim f(h) - f(0) = 0 h Therefore, lim f(h) - f(0) and so f’(0) exists and equals 0. h X ->0 + X ->0 - X ->0 + H -> 0

40. Bibliography http://www.mathwords.com/f/formula.htm http://images.google.com/imgres?imgurl=http://www.francis.edu/up loadedImages/Math/blackboard_math.gif&imgreful http://images.google.com/imgres?imgurl=http://www.francis.edu/up loadedImages/Math/blackboard_math.gif&imgreful www.open.salon.com www.babble.com www. school-clipart.com www.first90days.wordpress.com www.schools.sd68.bc.ca http://numericalmethods.eng.usf.edu/anecdotes/newton.html http://gardenofpraise.com/ibdnewt.htm http://scienceworld.wolfram.com/biography/leibniz/html http://calculusthemusical.com/wp- content/uploads/2008/02/matheatre-power-rule.mp3http://calculusthemusical.com/wp- content/uploads/2008/02/matheatre-power-rule.mp3 ©Copyright. Hunter Faulk. Megan Troutman. Anthony Stevens. February 19, 2010