DERIVATIVEFor y = f(x), the derivative of f at x, denoted by f(x), to be 𝒇 𝒙+ ∆𝒙 −𝒇(𝒙) 𝒅𝒚 𝒇 𝒙 = 𝐥𝐢𝐦 = = y’ ∆𝒙→𝟎 ∆𝒙 𝒅𝒙 if the limit exists.
𝑑𝑦 𝑑𝑥“derivative of y with respect to x” or “dee y overdee x”- means that the rate of change of y is based on the change on the value of x.TAKE NOTE:𝒅𝒚 is NOT beinng regarded as quotient, but as a𝒅𝒙single symbol.
Four-StepDifferentiation Process1. Replace x by 𝒙 + ∆𝒙 and y by 𝒚 + ∆𝒚.2. Solve for ∆𝒚 in terms of 𝒙 + ∆𝒙.3. Divide both sides by ∆𝒙. ∆𝑦4. Find the limit of as ∆𝒙 0. ∆𝑥
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 𝑑𝑥
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 𝑑𝑥
Rules in finding the DERIVATIVESThe Constant Multiple Rule If y = f(x) = 𝐶 ∗ 𝑓(𝑥), where f(x) is a differentiable function, then y’ = 𝐶 ∗ 𝑓′(𝑥) 𝑑𝑥 Also, = 𝐶 ∗ 𝑓′(𝑥) and f’(x) = 𝑑𝑦 ′ 𝐶∗ 𝑓 𝑥 .
Rules in finding the DERIVATIVESThe Sum and Difference Rule If y = f(x) = 𝑢 𝑥 ± 𝑣(𝑥)where u and v are differentiable functions, then y’ = u′(x) ± v’(x) 𝑑𝑦 Also, = 𝑑𝑥 ′ ′ ′ u x ± 𝑣 𝑥 𝑎𝑛𝑑 𝑢 𝑥 = 𝑢 ′ 𝑥 ± 𝑣′(𝑥)
Rules in finding the DERIVATIVESThe Power Rule 𝑛 If y = f(x) = 𝑥 , where x is a differentiable function and n is a real number, then y’ = 𝑛𝑥 𝑛−1 . 𝑑𝑦 𝑛−1 Also, = 𝑛𝑥 and f’(x) = 𝑑𝑥 𝑛−1 𝑛𝑥
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.
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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 ∗ 𝒗′(𝒙) + v(x) * u’(x). 𝒅𝒚 ′ Also, = 𝒖 𝒙 ∗ 𝒗 𝒙 + 𝒗 𝒙 ∗ 𝒅𝒙 ′ 𝒖′(𝒙) and f’(x) = 𝒖 𝒙 ∗ 𝒗 𝒙 + 𝒗 𝒙 ∗ 𝒖′(𝒙)
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 ∗𝒖′ 𝒙 −𝒖 𝒙 ∗𝒗′ (𝒙) y’ = [𝒗 𝒙 ] 𝟐 𝒗∗𝒖′ −𝒖∗𝒗′ Also, f’(x) = 𝟐 and 𝒗 𝒅𝒖 𝒅𝒗 𝒅𝒚 𝒗∗ −𝒖∗ 𝒅𝒙 𝒅𝒙 = 𝒅𝒙 𝒗𝟐
SEATWORK #1I. Find the derivatives of the following functions A. By applying the rules −𝟐𝟎 1. 𝒚 = 𝒙 𝟑 𝟓 2. 𝒚= 𝒙 𝟒 𝟐−𝟓𝒙 3. y= 𝟑𝒙−𝟏
𝟒 4. 𝒚 = −𝟓𝒙(𝒙 − 𝟒) 5. y = 3x 6. y = - 𝟏𝟖 𝟑 𝟐 7. y = 𝟔𝒙 − 𝟏𝟐𝒙 + 𝟕 𝒙 −𝟐 8. y = 𝒙𝟑B. using the four-process differentiation1. y = 𝟓𝒙 −10