General Differentiation Differentiation of Sine & Cosine Name ......Sec 2.5 – General Differentiation Differentiation of Sine & Cosine Name: (1) Let’s try to prove the derivative
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Sec 2.5 – General Differentiation Differentiation of Sine & Cosine Name:
(1) Let’s try to prove the derivative of the function 풇(풙) = 풔풊풏(풙) using the definition of the derivative.
풇′(풙) = 퐥퐢퐦풉→ퟎ
풔풊풏(풙+ 풉) − 풔풊풏(풙)
풉
We could use the sum and difference trig identities to substitute 풔풊풏(풙+ 풉) = 풔풊풏(풙)풄풐풔(풉) + 풄풐풔(풙)풔풊풏(풉)
풇′(풙) = 퐥퐢퐦풉→ퟎ
풔풊풏(풙)풄풐풔(풉) + 풄풐풔(풙)풔풊풏(풉) − 풔풊풏(풙)
풉
We can rearrange the terms of the numerator
풇′(풙) = 퐥퐢퐦풉→ퟎ
풔풊풏(풙)풄풐풔(풉) − 풔풊풏(풙) + 풄풐풔(풙)풔풊풏(풉)
풉
We could then factor out 풔풊풏(풙) from the first two terms.
풇′(풙) = 퐥퐢퐦풉→ퟎ
풔풊풏(풙)(풄풐풔(풉) − ퟏ) + 풄풐풔(풙)풔풊풏(풉)
풉
Next, we could use limit laws to rewrite the following statement:
풇 (풙) = 풔풊풏(풙) ∙ 퐥퐢퐦풉→ퟎ
(풄풐풔(풉) − ퟏ)
풉+ 풄풐풔(풙) ∙ 퐥퐢퐦
풉→ퟎ풔풊풏(풉)
풉
(A) Which leaves us with two indeterminate form limits. We will need to use the Squeeze Limit Theorem. Let’s first investigate the limit lim → . Consider using the following diagram for which 0 ≤ ℎ < .
푨풓풄푳풆풏품풕풉푩푫 ≤ 푩푪
풉 ≤ 퐭퐚퐧풉
풉 ≤ 퐬퐢퐧 풉퐜퐨퐬 풉
풉 ∙ 퐜퐨퐬 풉 ≤ 퐬퐢퐧 풉
퐜퐨퐬 풉 ≤ 퐬퐢퐧 풉풉
퐴푟푐퐿푒푛푔푡ℎ = 푟 ∙ 휃
퐿푒푛푔푡ℎ퐵퐷 = 1 ∙ ℎ
퐿푒푛푔푡ℎ퐵퐷 = ℎ tanℎ =
tanℎ = 퐵퐶
tan휃 =
푫푬 ≤ 푨풓풄푳풆풏품풕풉푩푫
퐬퐢퐧 풉 ≤ 풉
퐬퐢퐧 풉풉 ≤ 풉풉
퐬퐢퐧 풉풉 ≤ ퟏ
퐴푟푐퐿푒푛푔푡ℎ = 푟 ∙ 휃
퐿푒푛푔푡ℎ퐵퐷 = 1 ∙ ℎ
퐿푒푛푔푡ℎ퐵퐷 = ℎ sinℎ =
sinℎ = 퐷퐸
sin휃 =
퐜퐨퐬 풉 ≤ 퐬퐢퐧 풉풉≤ ퟏ
퐥퐢퐦풉→ퟎ
퐜퐨퐬 풉 ≤ 퐥퐢퐦풉→ퟎ
퐬퐢퐧 풉풉≤ 퐥퐢퐦
풉→ퟎퟏ
ퟏ ≤ 퐥퐢퐦풉→ퟎ
퐬퐢퐧 풉풉≤ ퟏ
퐥퐢퐦풉→ퟎ
퐬퐢퐧 풉풉 = ퟏ
By the Squeeze Limit Theorem:
퐥퐢퐦풉→ퟎ
퐬퐢퐧 풉풉
= 퐥퐢퐦풉→ퟎ
퐬퐢퐧( 풉)풉
sin(−푥) = − sin(푥)
Although we were only working with a right-handed limit we could find the left hand limit by substituting h with – h.
Since the function 푓(푥) = sin(푥) is an odd function