So the two points are insert the two points back into

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So, the two points are Insert the two points back into the y’ equation. One at a time. This tells us the slope (m) at the two points. Find the equation of the tangent lines at the two points. 41 Ch 2 AP Derivatives LP
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Example : Use implicit differentiation to show that if , then . 42 Ch 2 AP Derivatives LP
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Example : Show that, for the following relations, is as stated. Thus, find for each. a) b) 43 Ch 2 AP Derivatives LP
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c) Assignment : Red Book Page 107 1adg, 2ce, 3d, 5 Page 111 1afg, 2ad, 3, 5, 7c, 10 44 Ch 2 AP Derivatives LP
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Math 31AP Unit: Derivatives Day: 6 Topic: Derivatives of Sine and Cosine Functions Quiz on Differentiation Techniques to Start Class Trig Identities to Know Reciprocal Quotient Pythagorean Double Angle Trig Ratios to Know Chart in Radians Derivatives of Trig Functions Example : Find from first principles the derivatives of a. b. Example : Differentiate each of the following. a) b) c) d) e) f) 45 Ch 2 AP Derivatives LP
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Example : Find the equation of the line tangent to the curve when . . FROM SALISBURY NOTES Derivative of y = sin x therefore, Derivative of y = cos x y = cosx * y = y / = 1 y / = = cosx(0) – sinx(1) y / = sinx If y = sinx; y / = cosx and if y = cosx; y / = sinx ( note slopes of graphs) 46 Ch 2 AP Derivatives LP
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eg 1. y = 5sinx y / = 5cosx 2. y = cos3x y / = sin3x 3 = 3sin3x 3. y = 2sin( 4x) y / = 2cos( 4x) 4 = 8cos( 4x) = 8cos(4x) even function 4. y = sin 2 x = (sin x) 2 y / = or sin2x 5. y = sin 3 (x 2 + 3) = [sin(x 2 + 3)] 3 y / = 3sin 2 (x 2 + 3)cos(x 2 + 3)(2x) y / = 6xsin 2 (x 2 + 3)cos(x 2 + 3) 6. y = cosxsinx OR y = cosxsinx = y / = sinxsinx + cosxcosx y / = sin 2 x + cos 2 x = cos2x 7. y = cos(sinx 2 ) y / = sin(sinx 2 )(cosx 2 )(2x) y / = 2xsin(sinx 2 )(cosx 2 ) 8. y = x(cos 2 x + 1) OR y = xcos 2 x + x y / = cos 2 x + 1 + 2cosx( sinx)(x) y / = cos 2 x + 2cosx( sinx)(x) + 1 y / = cos 2 x + 1 2xcosxsinx y / = cos 2 x 2xcosxsinx + 1 or y / = cos 2 x + 1 xsin2x 9. y = sin 2 x + cos 2 x OR y = 1 y / = 2sinxcosx + 2cosx( sinx) y / = 0 y / = 0 10.y = tanx y = sinx(cosx) 1 y / = cosx(cosx) 1 (cosx) 2 ( sinx)(sinx) y / = 1 + (sinx) 2 (cosx) 2 y / = (cosx) 2 [(cosx) 2 + (sinx) 2 ] y / = or sec 2 x 11.y = cotx = (tanx) 1 or y = cotx 47 Ch 2 AP Derivatives LP
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y / = (tanx) 2 (sec 2 x) 12.y = sinxcosy 13.y 2 + tany = 3x 2yy / + sec 2 y(y / ) = 3 14.y + ycosx = 3 * y / + y / cosx sinx(y) = 0 y / (1 + cosx) = ysinx Assignment Read 308 – 313 pg. 313 # 1 a – f, h, j, k, m, o 2 a, b, c 3 a 4a (Salisbury) Assignment : Red Book Page 313 1aegilnpt, 2aef, 3bd (AP Notes) 48 Ch 2 AP Derivatives LP
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Additional Questions 1. y = sin 2 (x 3 ) ANS: y / = 6x 2 sin(x 3 )cos(x 3 ) or 3x 2 sin(2x 3 ) 2. y = 4sinxcosx ANS: y / = 4cos(2x) 3. y = cos(cosx) ANS: y / = sinxsin(cosx) 4. y = cos(x + 3) 2 ANS: y / = 2(x + 3)sin(x + 3) 2 5. y = cos 2 x sin 2 x ANS: y / = 2sin(2x) 6. y = sin(2x)cos(4x) + cos(2x)sin(4x) ANS: y = sin(6x) y / = 6cos(6x) 49 Ch 2 AP Derivatives LP
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Math 31AP Unit: Derivatives Day: 7 Topic: Derivatives of Other Trig Functions Derivatives of Other Trig Ratios Example : Use the identities and rules of differentiation to find the derivatives of a. b. c. d. Proofs of 'a', 'b', 'c' and 'd'. a. b. y = csc(x) = (sinx) 1 c. y = sec(x) = (cosx) 1 y / = (sinx) 2 (cosx) y / = (cosx) 2 ( sinx) d. 50 Ch 2 AP Derivatives LP
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Example : Differentiate each of the following. a. . b. ANSWER: . c. ANSWER: . 51 Ch 2 AP Derivatives LP
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d. ANSWER: 52 Ch 2 AP Derivatives LP
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X Y Example : Find the equation of the line tangent to the curve when .
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