Math RE - Calculus I Trigonometry Limits & Derivatives Page 1 of 8. x = 1 cos x. cos x 1 = lim

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1 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page of 8 Trigonometric Limits It has been shown in class that: lim 0 sin lim 0 sin lim 0 cos cos 0 lim 0 cos lim 0 + cos + To evaluate trigonometric limits, use these results and the properties of limits shown in another tutorial. Eamples: lim cos sec cos sec lim 0 lim 0 sin sin 4 Evaluate: lim 0 lim 0 lim 0 sin sin cos 5 Evaluate: lim 0 cos 0 cos 0 6 Evaluate: lim 0 sin lim 0 cos + cos cos lim lim + cos lim sin sin lim 0 sin lim 0 sin 0. 0 Trigonometric Derivatives formulas it has been shown in class that: sin cos ; cos sin ; tan sec csc csc cot ; sec sec tan ; cot csc If u is a function of, then the derivatives are: sinu cosu.u ; cosu sinu.u ; tanu sec u.u cscu cscu cotu.u ; secu secu tanu.u ; cotu csc u.u

2 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page of 8 Trigonometric Derivative basic a Find the derivative of y cos 5csc sin 5 csc cot sin + 5 csc cot d b Evaluate y y sin + 5 csc cot + 50 Trigonometric Derivative product a Find the derivative of y sec + sin Use product rule: u.v u.v+ v.u u sec u sec tan ; v +sin v cos sec tan + sin + cos sec sec tan +sec tan sin + d y sec tan + tan +sec tan +sec sec tan +sec b Evaluate y y sec tan + sec 0 Trigonometric Derivative quotient a Find the derivative of y Use quotient rule: u v cos sin + u.v v.u u cos u sin ; v sin + v cos v sin sin + cos cos sin sin + cos cos d sin + sin + y sin sin cos sin + y sin + sin + sin + b Evaluate y 0 y 0 sin +cos sin sin sin + sin + sin 0 +

3 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page of 8 Trigonometric Derivative composition a Find the derivative of y tan sin Use chain rule: uv u v.v uv tan sin u v sec sin ; v sin v cos d sec sin. cos b Evaluate y 0 y 0 sec sin. cos sec Trigonometric Derivative implicit a Find y if sin y y + y cos use implicit differentiation: sin y product rule: sin y + y sin y y y y cos product rule: y cos + y sin Derivative equation: sin y + y sin y y + y cos y sin y sin y y y cos y sin sin y y sin y ++cos y sin +siny y y sin +siny + cos sin y b Evaluate y, y, sin +sin +cos sin 0+ 0

4 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page 4 of 8 Trigonometric Derivative product with eponential Use the eponential derivatives shown in another tutorial. a Find the derivative of y sin e Use product rule: u.v u.v+ v.u u sin u cos ; v e v e d cos e +e sin e cos + sin b Evaluate y 0 y 0 e 0 cos0 + sin0 Trigonometric Derivative quotient with eponential a Find the derivative of y Use quotient rule: u v cos e u.v v.u u cos u sin ; v e v e v d sine e cos e e sin +cos sin +cos e e b Evaluate y 0 y 0 sin 0 + cos e 0 Trigonometric Derivative composition with eponential a Find the derivative of y e cos Use chain rule: uv u v.v uv e cos u v e cos ; v cos v sin d e cos. sin sine cos b Evaluate y y sin e cos e0

5 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page 5 of 8 Trigonometric Derivative implicit with eponential a Find y if e y cos tan y cosy + e use implicit differentiation: e y cos product rule: y e y cos +e y sin tan y y sec y cos y + e product rule: y sin y + e Derivative equation: y e y cos +e y sin y sec y y sin y + e y e y cos y sec y + y sin y e y sin + e y e y cos sec y +siny e y sin + e y e y sin + e e y cos sec y +siny b Evaluate y 0, 0 y 0, 0 e 0 sin 0 + e 0 e 0 cos 0 sec 0 + sin Trigonometric Derivative product with logarithm Use the logarithm derivatives shown in another tutorial. a Find the derivative of y cos ln Use product rule: u.v u.v+ v.u u cos u sin ; v ln v sin ln + d cos b Evaluate y 0 y 0 sin0 ln + cos0

6 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page 6 of 8 Trigonometric Derivative quotient with logarithm a Find the derivative of y Use quotient rule: u v ln + +sin u.v v.u u ln + u + ; v +sin v cos d + v + sin cos ln + + sin b Evaluate y 0 y 0 + sin 0 cos 0 ln + sin 0 Trigonometric Derivative composition with logarithm a Find the derivative of y ln4cos Use chain rule: uv u v.v uv ln 4 cos u v d 4 sin. 4 sin 4cos 4cos 4cos ; v 4 cos v 4 sin b Evaluate y y 4sin 4cos 4 4

7 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page 7 of 8 Trigonometric Derivative implicit with logarithm a Find y if sin +lnsiny ln cos + + cos y + use implicit differentiation: sin +lnsiny cos + cos y sin y y cos + y cot y ln cos + + cos y + Derivative equation: sin cos + y sin y cos + y cot y y cot y + y sin y y cot y +siny y sin cos + y sin y sin cos cos + cos + cot y +siny sin sin cos cos + cos cos + cos + sin cos cos + cos + b Evaluate y, y, sin cos cos cos + + cot +sin 0 Trigonometric Derivative with eponential & logarithm a Find the derivative of y e sin +cosln d cos e sin + sin ln cos e sin sin ln b Evaluate y y cos e sin sin ln e 0 sin 0

8 Math 0-0-RE - Calculus I Trigonometry Limits & Derivatives Page 8 of 8 Trigonometric Derivative with eponential & logarithm a Find the derivative of y lne +cos Use chain rule: uv u v.v uv lne +cos u v d e +cos. e sin e sin e +cos e +cos ; v e +cos v e sin b Evaluate y 0 y 0 e0 sin 0 e 0 + cos 0 Trigonometric Derivative with logarithmic differentiation a Find the derivative of y + cos Use logarithmic differentiation: Take ln on both sides lny ln + cos ; use log properties: lny cos ln + implicit differentiation on Left-Hand-Side and product rule on Right-Hand-Side: y y sin ln ++ + cos y y [ sin ln ++ cos ] [ + cos sin ln ++ cos ] + + b Evaluate y 0 [ y 0 cos0 sin0 ln + cos0 ]