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The Basics of Derivatives

The Basics of Derivatives. Table of Contents. Definition. Chain Rule. Directions. Basic. Natural Log. Product Rule. Trig. Quiz. Quotient Rule. More Examples. Directions. Question #?. Return to Table of Contents . Next Slide. Go to specific quiz question. Quiz.

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The Basics of Derivatives

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  1. The Basics of Derivatives

  2. Table of Contents Definition Chain Rule Directions Basic Natural Log Product Rule Trig Quiz Quotient Rule MoreExamples

  3. Directions Question #? Return to Table of Contents Next Slide Go to specific quiz question Quiz Previous Slide Return to Quiz Main Page The correct order for going through this PowerPoint Presentation is to start at the right-hand side and go down through the items. Then go to the top of the left-hand side. Save the quiz for last! Each category will link you to the category that comes after it.

  4. Definition What is a Derivative? The Dictionary defines a derivative as the: “limit of the ratio of the increment of a function to the increment of a variable in it, as the latter tends to 0; the instantaneous change of one quantity with respect to another.” Quiz!

  5. Definition A derivative is the slope of the original function. The original function is y=x2-3 or the blue line The slope or derivative is y=2x, the red line Quiz!

  6. Definition The notation for a derivative can be written in many different ways, all of which are correct: f’(x) y’ Quiz!

  7. Definition Formula of a derivative: Quiz!

  8. Definition Example: continued… Quiz!

  9. Definition Question #2 Question #9 Quiz!

  10. Basic Derivative a and b are constants If given: y=ax+b then: y’=a For the equation of a line, the derivative is always the slope of the line given: y=a then: y’=0 Whenever you take the derivative of a constant number, the answer will always be zero. a is a constant Quiz!

  11. Basic Derivatives When you have x to a power, bring the power down and decrease the power of x by one. Quiz!

  12. Basic Derivatives Miscellaneous Rules: Constant Multiple Rule Sum and Difference Rules Quiz!

  13. Basic Derivatives Examples: Question #1 Quiz!

  14. Product Rule Definition: Given: y= uv Then: y’= vu’ + uv’ Quiz!

  15. Product Rule Does it matter if you switch the equation around making it uv’ + vu’ instead of vu’ + uv’ ??? Either way you do it, you will get the same answer for the derivative. NO! Quiz!

  16. Product Rule “It’s the FIRST times the derivative of the SECOND plus the SECOND times the derivative of the FIRST.” Catchy Saying To Remember Product Rule: Quiz!

  17. Product Rule Example: Question #3 Quiz!

  18. Quotient Rule Definition: Quiz!

  19. Quotient Rule Does it matter if you switch the variables around as you can do for the Product Rule??? Since the numerator is subtraction, switching the variables does make a difference in the answer. YES!!! Quiz!

  20. Quotient Rule Example: Question #4 Quiz!

  21. Chain Rule Definition: Quiz!

  22. Chain Rule Example: Question #6 Question #10 Quiz!

  23. Natural Log Definition: Quiz!

  24. Natural Log Example: Quiz!

  25. Trig Problems For Trig problems you must memorize the derivative of 6 trig functions that will then make it easier to do the harder problems. Quiz!

  26. Trig Problems Commit these to memory: Given: y= sinx Given: y= cscx y’= cosx y’= -cscxcotx Given: y= cosx Given: y= secx y’=-sinx y’= secxtanx Given: y= tanx Given: y= cotx y’= sec2x y’= csc2x Quiz!

  27. Trig Problems Tricks to memorizing the 6 trig derivatives: • For all the functions that begin with a ‘c’, the derivative will be negative. • The following have similar derivatives: • Cscx and secx • Tanx and cotx • Sinx and cosx Quiz!

  28. Trig Problems Example: Given: y= cotx + tanx + sinx y’= -csc2x + sec2x + cosx Quiz!

  29. Trig Problems Example: Question #7 Question #8 Quiz!

  30. Examples The examples that will be shown are a little tougher than those in the rest of the slides. Don’t worry if you are struggling with them. We will go over them in class. Quiz!

  31. Examples Combining the Chain Rule and the Product Rule: Quiz!

  32. Examples Chain rule + quotient rule Quiz!

  33. Examples Chain rule + Quotient Rule continued… Quiz!

  34. Example Natural Log problem with Product Rule: Quiz!

  35. Applying Derivatives To Real Life • The Derivative of the position of an object is its instantaneous velocity. • The Derivative of an objects instantaneous velocity is its average velocity. and Question #5 Quiz!

  36. Quiz Time!!! • Question 6 • Question 7 • Question 8 • Question 9 • Question 10 • Question 1 • Question 2 • Question 3 • Question 4 • Question 5

  37. Question 1 • Find the Derivative of y= -2x + 4 A. -2x + 4 B. -2 + x C. -2 D. 2 E. 4 Quiz!

  38. Good guess, but that is not the correct answer. Try Again! Click for more review Quiz!

  39. FANTASTIC!!! That is the correct answer! Way to go! Next Question Quiz!

  40. Question 2 2. Which of the following statements are correct? A. A derivative is the perpendicular line to a function B. The derivative is the slope of the function C. A derivative can be expressed as D. All of the above E. B and C Quiz!

  41. Good guess, but that is not the correct answer. Try Again! Click for more review Quiz!

  42. FANTASTIC!!! That is the correct answer! Way to go! Quiz! Next Question

  43. Question 3 3. Find the derivative of y= xlnx A. B. x C. 1 + lnx D. Quiz!

  44. Good guess, but that is not the correct answer. Try Again! Click for more review Quiz!

  45. FANTASTIC!!! That is the correct answer! Way to go! Quiz! Next Question

  46. Question 4 4. What rule is shown: A. Quotient Rule B. Chain Rule C. Product Rule D. L’Hospital’s Rule Quiz!

  47. Good guess, but that is not the correct answer. Try Again! Click for more review Quiz!

  48. FANTASTIC!!! That is the correct answer! Way to go! Quiz! Next Question

  49. Question 5 5. True or False: The derivative of an object’s instantaneous velocity is its position. A. True B. False Quiz!

  50. Good guess, but that is not the correct answer. Try Again! Click for more review Quiz!

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