Unfortunately the derivative of a product is not the product of the derivatives. For example, if

Transcription

1 Prodct Rle Unortnately te deriatie o a prodct is not te prodct o te deriaties. For eample, i Ten p So is p bt 11 1, and tey are not eal in general. Tat [ is not ] in general To compte te deriatie o a prodct let p, were te two nctions and are bot dierentiable. Using te it deinition: p p p MATH 18 Lectre 11 1 o 17 Ronald Brent 18 All rigts resered.

2 MATH 18 Lectre 11 o 17 Ronald Brent 18 All rigts resered. By sbtracting and adding we can write p Wic can be rewritten as p. Eialently

3 Eamples: a 1 1, ten sing te prodct rle Here 1 and 1, so and 1 1 I we simpliy tis, we get 5. Tere are some nctions oweer tat can t be simpliied. Tey sally inole trig and eponential nctions. MATH 18 Lectre 11 o 17 Ronald Brent 18 All rigts resered.

4 5 b sin, 5 Here and sin, so 5 and cos 5 5 sin cos 5sin cos 8 c sin, 8 7 sin 8 cos MATH 18 Lectre 11 o 17 Ronald Brent 18 All rigts resered.

5 d cos, cos sin cos sin 1 1 e g t t cost, g t cost t sin t cost t sin t t t sin, 1 sin cos g sin cos cos cos sin sin cos sin MATH 18 Lectre 11 5 o 17 Ronald Brent 18 All rigts resered.

6 MATH 18 Lectre 11 6 o 17 Ronald Brent 18 All rigts resered. Qotient Rle To determine te otient rle, let, were and are bot dierentiable nctions and. Using te it deinition, Sbtracting and adding in te nmerator gies

7 MATH 18 Lectre 11 7 o 17 Ronald Brent 18 All rigts resered. wic can be rewritten as So tat

8 or Eamples: a I, Here and, so 1 and 1 MATH 18 Lectre 11 8 o 17 Ronald Brent 18 All rigts resered.

9 b I, 1 Here and 1, so and MATH 18 Lectre 11 9 o 17 Ronald Brent 18 All rigts resered.

10 c I, sin sin sin cos sin sin cos Alternatiely we cold bring te sin p as a csc. sin csc csc csc cot 1 cos sin sin sin sin cos sin sin sin sin sin cos Tis is a case wen te otient rle is deinitely better! MATH 18 Lectre 11 1 o 17 Ronald Brent 18 All rigts resered.

11 d I , e sin 1 cos 1 cos cos sin 1 cos sin cos cos sin 1 cos cos 1 1 cos 1 1 cos MATH 18 Lectre o 17 Ronald Brent 18 All rigts resered.

12 Deriaties o te Remaining Trigonometric Fnctions sin cos 1 1 Since tan, cot, sec, and csc, we can se te cos sin cos sin otient rle to compte te deriaties o tese nctions. cos cos sin sin cos sin 1 tan sec cos cos cos cot sin sin cos cos cos sin 1 csc sin sin sin cos 1 sin sin sec cos cos 1 sin cos cos sec tan sin 1cos cos csc sin sin 1 cos sin sin csc cot MATH 18 Lectre 11 1 o 17 Ronald Brent 18 All rigts resered.

13 To smmarize: sin cos cos sin tan sec cot csc sec sec tan csc csc cot Eamples: a sec sec sec tan sec tan b cot 6 cot csc c sin tan cos tan sin sec sin 1 sec d csc csc csc cot csc cot cos sin cos 1 sin cos e 1 or yo cold write cos and se te prodct rle. Try it. MATH 18 Lectre 11 1 o 17 Ronald Brent 18 All rigts resered.

14 tan sec 1 sec 1sec tan sec tan sec 1 1 sec sec sec tan sec 1 sec sec sec sec 1 tan sec sec 1 sec 1 sec sec 1 g sec tan sec tan tan sec sec sec tan sec MATH 18 Lectre 11 1 o 17 Ronald Brent 18 All rigts resered.

15 sin 1 1cos sin 1 cos cos sin 1 i cos tan 1 tan 1cos sin tan 1 cos sec j sec tan sec tan sec tan sec tan sec MATH 18 Lectre o 17 Ronald Brent 18 All rigts resered.

16 Eample: Find te eation o te line tangent to te grap o y tan at te point π, First compte te slope. m y sec 1. So Plgging in te point gies π π y b π b and so b π. And so y π π Eample: Find te line tangent to te grap o y cot at te point, 1 m y First compte te slope. csc. So Plgging in te point gies π / π / y b π 1 b and so b 1 π And so π y 1 MATH 18 Lectre o 17 Ronald Brent 18 All rigts resered.

17 Eample: Dierentiate sin tan sin tan sin tan d sin tan d cos tan sin sec sin tan sin sin sec sin [ tan 1 sec ] Eample: Dierentiate cot 1 cot 1 csc cot 1 cot 1 cot 1 csc MATH 18 Lectre o 17 Ronald Brent 18 All rigts resered.