Differentiation

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Differentiation

  1. 1. DERIVATIVEFor y = f(x), the derivative of f at x, denoted by f(x), to be 𝒇 𝒙+ βˆ†π’™ βˆ’π’‡(𝒙) π’…π’š 𝒇 𝒙 = π₯𝐒𝐦 = = y’ βˆ†π’™β†’πŸŽ βˆ†π’™ 𝒅𝒙 if the limit exists.
  2. 2. 𝑑𝑦 𝑑π‘₯β€œderivative of y with respect to x” or β€œdee y overdee x”- means that the rate of change of y is based on the change on the value of x.TAKE NOTE:π’…π’š is NOT beinng regarded as quotient, but as a𝒅𝒙single symbol.
  3. 3. Four-StepDifferentiation Process1. Replace x by 𝒙 + βˆ†π’™ and y by π’š + βˆ†π’š.2. Solve for βˆ†π’š in terms of 𝒙 + βˆ†π’™.3. Divide both sides by βˆ†π’™. βˆ†π‘¦4. Find the limit of as βˆ†π’™ οƒ  0. βˆ†π‘₯
  4. 4. Examples1.y = 2x – 3
  5. 5. 2. y = 4x - 12
  6. 6. 3. y = x - 2
  7. 7. Increment Method1.Evaluate 𝑓 π‘₯ + β„Ž .2.Subtract by 𝑓 π‘₯ .3.Divide by h.4.Find the limit as h οƒ  0
  8. 8. Examples1. y = 2x – 3
  9. 9. 2. y = 4x - 12
  10. 10. 3. y = x - 2
  11. 11. Rules in finding the DERIVATIVESThe Constant Function Rule If y = f(x) = C, where C is a constant, then y’ = 0. 𝑑𝑦 Also, = 0 and f’(x) = 0 𝑑π‘₯
  12. 12. Rules in finding the DERIVATIVESThe Identity Function Rule If y = f(x) = x, where x is a differentiable function, then y’ = 1. 𝑑𝑦 Also, = 1 and f’(x) = 1 𝑑π‘₯
  13. 13. Rules in finding the DERIVATIVESThe Constant Multiple Rule If y = f(x) = 𝐢 βˆ— 𝑓(π‘₯), where f(x) is a differentiable function, then y’ = 𝐢 βˆ— 𝑓′(π‘₯) 𝑑π‘₯ Also, = 𝐢 βˆ— 𝑓′(π‘₯) and f’(x) = 𝑑𝑦 β€² πΆβˆ— 𝑓 π‘₯ .
  14. 14. Rules in finding the DERIVATIVESThe Sum and Difference Rule If y = f(x) = 𝑒 π‘₯ Β± 𝑣(π‘₯)where u and v are differentiable functions, then y’ = uβ€²(x) Β± v’(x) 𝑑𝑦 Also, = 𝑑π‘₯ β€² β€² β€² u x Β± 𝑣 π‘₯ π‘Žπ‘›π‘‘ 𝑒 π‘₯ = 𝑒 β€² π‘₯ Β± 𝑣′(π‘₯)
  15. 15. Rules in finding the DERIVATIVESThe Power Rule 𝑛 If y = f(x) = π‘₯ , where x is a differentiable function and n is a real number, then y’ = 𝑛π‘₯ π‘›βˆ’1 . 𝑑𝑦 π‘›βˆ’1 Also, = 𝑛π‘₯ and f’(x) = 𝑑π‘₯ π‘›βˆ’1 𝑛π‘₯
  16. 16. FLASH IT! (INDIVIDUAL TASK) Some students find it hard to memorizethe different rules in differentiation. In thisperformance task, you are to make at least 5flash cards involving differentiation rules. Inthis flash card, you need to put in all rules indifferentiation (Sum rule, Constant Multiplerule, etc.) Take note that, in front of your flashcard you must state the rule for differentiationand on its back, write at least 3 examples. Youwill be graded according to content, creativityand punctuality. Put your flash cards in anenvelope or anything that will keep your flashcards together.
  17. 17. Anchor Good(5) Adequate(3) Poor(1) Weight Score Cards contain Cards contain Cards contain rules for rules for rules for differentiations Content differentiation differentiations 5 25 were some and examples were examples examples are not were correct. are not correct. correct. The cards are The cards are The cards are not presentable and presentable and presentable andCreativity somewhat 3 15 were colourful not colourful and colourful and and neat. neat. neat. There were less There were than 5 cards and The fan did not atleast 5 cards.Compliance were somewhat comply with the 3 15 and were colorful presented size. and neat creatively. The cards were The cards were The cards were Date of submitted on the submitted a day submitted 2 days 2 10submission day of after the date of after the date of submission submission submission Total 70
  18. 18. Rules in finding the DERIVATIVESThe Product Rule If y = f(x) = u(x) * v(x), where u and v are differentiable functions, then y’ = 𝒖 𝒙 βˆ— 𝒗′(𝒙) + v(x) * u’(x). π’…π’š β€² Also, = 𝒖 𝒙 βˆ— 𝒗 𝒙 + 𝒗 𝒙 βˆ— 𝒅𝒙 β€² 𝒖′(𝒙) and f’(x) = 𝒖 𝒙 βˆ— 𝒗 𝒙 + 𝒗 𝒙 βˆ— 𝒖′(𝒙)
  19. 19. Rules in finding the DERIVATIVESThe Quotient Rule 𝒖(𝒙) If y = f(x) = , where u and v 𝒗(𝒙) are differentiable function then, 𝒗 𝒙 βˆ—π’–β€² 𝒙 βˆ’π’– 𝒙 βˆ—π’—β€² (𝒙) y’ = [𝒗 𝒙 ] 𝟐 π’—βˆ—π’–β€² βˆ’π’–βˆ—π’—β€² Also, f’(x) = 𝟐 and 𝒗 𝒅𝒖 𝒅𝒗 π’…π’š π’—βˆ— βˆ’π’–βˆ— 𝒅𝒙 𝒅𝒙 = 𝒅𝒙 π’—πŸ
  20. 20. SEATWORK #1I. Find the derivatives of the following functions A. By applying the rules βˆ’πŸπŸŽ 1. π’š = 𝒙 πŸ‘ πŸ“ 2. π’š= 𝒙 πŸ’ πŸβˆ’πŸ“π’™ 3. y= πŸ‘π’™βˆ’πŸ
  21. 21. πŸ’ 4. π’š = βˆ’πŸ“π’™(𝒙 βˆ’ πŸ’) 5. y = 3x 6. y = - πŸπŸ– πŸ‘ 𝟐 7. y = πŸ”π’™ βˆ’ πŸπŸπ’™ + πŸ• 𝒙 βˆ’πŸ 8. y = π’™πŸ‘B. using the four-process differentiation1. y = πŸ“π’™ βˆ’10
  22. 22. C. Using the increment method2. y = 18x + 2

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