121616949-math.85

# 121616949-math.85 - 1 sin x cos x ⇒ 2 sin(cos x ⇒ 3 √...

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4.5 Derivatives of the Trigonometric Functions 71 2. x sin x 3. 1 sin x 4. x 2 + x sin x 5. p 1 - sin 2 x All of the other trigonometric functions can be expressed in terms of the sine, and so their derivatives can easily be calculated using the rules we already have. For the cosine we need to use two identities, cos x = sin( x + π 2 ) , sin x = - cos( x + π 2 ) . Now: d dx cos x = d dx sin( x + π 2 ) = cos( x + π 2 ) · 1 = - sin x d dx tan x = d dx sin x cos x = cos 2 x + sin 2 x cos 2 x = 1 cos 2 x = sec 2 x d dx sec x = d dx (cos x ) - 1 = - 1(cos x ) - 2 ( - sin x ) = sin x cos 2 x = sec x tan x The derivatives of the cotangent and cosecant are similar and left as exercises. Exercises Find the derivatives of the following functions. 1. sin x cos x
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Unformatted text preview: 1. sin x cos x ⇒ 2. sin(cos x ) ⇒ 3. √ x tan x ⇒ 4. tan x/ (1 + sin x ) ⇒ 5. cot x ⇒ 6. csc x ⇒ 7. x 3 sin(23 x 2 ) ⇒ 8. sin 2 x + cos 2 x ⇒ 9. sin(cos(6 x )) ⇒ 10. Compute d dθ sec θ 1 + sec θ . ⇒ 11. Compute d dt t 5 cos(6 t ). ⇒ 12. Compute d dt t 3 sin(3 t ) cos(2 t ) . ⇒ 13. Find all points on the graph of f ( x ) = sin 2 ( x ) at which the tangent line is horizontal. ⇒ 14. Find all points on the graph of f ( x ) = 2 sin( x )-sin 2 ( x ) at which the tangent line is horizontal. ⇒...
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