Ch 2 Derivatives Notes

# So the two points are insert the two points back into

• Notes
• 85

This preview shows page 41 - 54 out of 85 pages.

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 Subscribe to view the full document.

Example : Use implicit differentiation to show that if , then . 42 Ch 2 AP Derivatives LP Example : Show that, for the following relations, is as stated. Thus, find for each. a) b) 43 Ch 2 AP Derivatives LP Subscribe to view the full document.

c) Assignment : Red Book Page 107 1adg, 2ce, 3d, 5 Page 111 1afg, 2ad, 3, 5, 7c, 10 44 Ch 2 AP Derivatives LP 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 Subscribe to view the full document.

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 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 Subscribe to view the full document.

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 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 Subscribe to view the full document.

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 Example : Differentiate each of the following. a. . b. ANSWER: . c. ANSWER: . 51 Ch 2 AP Derivatives LP Subscribe to view the full document.

d. ANSWER: 52 Ch 2 AP Derivatives LP X Y Example : Find the equation of the line tangent to the curve when . Subscribe to view the full document. {[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern