Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
UNIT 3 Lesson 3 Derivatives of Sine and Cosine 1 DERIVATIVE OF THE SINE FUNCTION π = π¬π’π§ π π π =? π π 2 Find the derivative of f(x) = sin x from first principles: π¬π’π§ π + π β π¬π’π§ π πβπ π+π βπ πβ² π = π₯π’π¦ Sum Identity: sin (A + B) = sin A cos B + cos A sin B π¬π’π§ π ππ¨π¬ π + ππ¨π¬ π π¬π’π§ π β π¬π’π§ π πβπ π πβ² π = π₯π’π¦ π¬π’π§ π ππ¨π¬ π β π¬π’π§ π + ππ¨π¬ π π¬π’π§ π πβπ π πβ² π = π₯π’π¦ π¬π’π§ π (ππ¨π¬ π β π)) + ππ¨π¬ π π¬π’π§ π πβπ π πβ² π = π₯π’π¦ πβ² π¬π’π§ π (ππ¨π¬ π β π)) ππ¨π¬ π π¬π’π§ π π = π₯π’π¦ + π₯π’π¦ πβπ π πβπ π ππ¨π¬ π β π π¬π’π§ π + π₯π’π¦ ππ¨π¬ π π₯π’π¦ πβπ πβπ π‘βπ π π πβ² π = π₯π’π¦ π¬π’π§ π × π₯π’π¦ πβπ continued 3 ππ¨π¬ π π¬π’π§ π π π = π¬π’π§ π π₯π’π¦ + ππ¨π¬ π π₯π’π¦ πβπ π πβπ π β² From the limits that we developed in the last lesson, we know that ππ¨π¬ π β π π¬π’π§ π π₯π’π¦ =π π₯π’π¦ =π πβπ πβπ π π πβ² π = π¬π’π§ π π + ππ¨π¬ π(π) πβ² π = ππ¨π¬ π Our first trigonometric derivative is: π π¬π’π§ π = ππ¨π¬ π π π 4 EXAMPLE 1: Find the derivative of π π = π π¬π’π§ π πβ² π = π ππ¨π¬ π πππ π½ = π =π π πππ π½ = π =π π Find the slope of the tangent at πππ π½ = , π π= π π π= π π π π π π = π ππ¨π¬ =π = π π π π π π β² π = π ππ¨π¬ = π(π) = π π π π=π πβ² π = π ππ¨π¬ π = π βπ = βπ π π β² 5 EXAMPLE 2: Find the derivative of π π = π¬π’π§(ππ) Use the Chain Rule πβ² π = (ππ¨π¬(ππ))(π) = π ππ¨π¬(ππ) Find the slope of the tangent at π=π πβ² π = π ππ¨π¬ π × π = π cos π = π π = π π π= π πβ² π π π = π ππ¨π¬ π × = π ππ¨π¬ = π π = π π π π π π= π πβ² π π = π ππ¨π¬ π × = π ππ¨π¬ π = π βπ = βπ π π 6 EXAMPLE 3: Differentiate h(x) = β 2sin(5x) πβ²(π) = βπ ππ¨π¬(ππ)(π) πβ²(π) = βππ ππ¨π¬(ππ) . Find the slope of the tangent at π π= π π π πβ² = βππ ππ¨π¬ π × = βππ ππ¨π¬ π = βππ βπ = ππ π π 7 EXAMPLE 4: Find the derivative of π = π¬π’π§ π π Find the slope of the tangent at π=π π π = π π¬π’π§ π ππ¨π¬ π π π π π = π π¬π’π§ π ππ¨π¬ π π π π π = π βπ π = π π π 8 EXAMPLE 5: Find the derivative of π = π π¬π’π§π π = π π¬π’π§ π π π = π π¬π’π§ π ππ¨π¬ π π π Find the slope of the tangent at π=π π π¬π’π§ π ππ¨π¬ π = π π βπ = π 9 π EXAMPLE 6: Find the derivative of π¦ = sin2 3π₯ = sin(3π₯) 2 π π = π π¬π’π§(ππ)(π) ππ¨π¬(ππ) π π π π = π π¬π’π§(ππ) ππ¨π¬(ππ) π π Find the slope of the tangent at π π π π π π= = π π¬π’π§(π × ) ππ¨π¬(π × ) ππ π π ππ ππ π π π π = π π¬π’π§ ππ¨π¬ π π π π π π π π× × = =π π π π 10 EXAMPLE 7: Differentiate π π₯ = 4 sin2 π₯ 2 + 2 π π = π ππππ ππ + π π π π = π π¬π’π§ π + π REWRITE π Using the Chain Rule β πβ² π = π(π) π¬π’π§ ππ + π ππ¨π¬ ππ + π (ππ) πβ² π = πππ π¬π’π§ ππ + π ππ¨π¬ ππ + π Using the sine double angle identity 2sin A cos A = sin 2A πβ² π = ππ π π¬π’π§ ππ + π ππ¨π¬ ππ + π πβ² π = ππ π¬π’π§ π ππ + π 11 DERIVATIVE OF THE COSINE FUNCTION What is d cos x dx ? 12 To find the derivative of the function f(x) = cos x, we can use a complementary angle identity in combination with the derivative of f(x) = sin x. COMPLEMENTARY ANGLES (sum is A 90o or π ) π c b B C a Angles A and B are complementary angles. π π A= βB and B = β A π π π π π ππ¨π¬ π© = = π¬π’π§ π¨ = = π¬π’π§( β π©) π π π π π π π¬π’π§ π¨ = = πππ π© = = πππ( β π¨) π π π 13 Find the derivative of π = ππ¨π¬ π π π = ππ¨π¬ π = π¬π’π§ βπ π We now differentiate using the Chain Rule: ππ¦ π = cos β π₯ (β1) ππ₯ 2 ππ¦ π = βcos β π₯ ππ₯ 2 ππ¦ = β sin π₯ ππ₯ This is the second trigonometric derivative: π ππ¨π¬ π = β π¬π’π§ π π π 14 EXAMPLE 8: Differentiate f(x) = sin (x) cos (x) and use the derivative to find the slope of the tangent to the function when π π₯= 3 Using the Product Rule πβ² π = π¬π’π§ π β π¬π’π§ π + ππ¨π¬ π ππ¨π¬ π 2 3 πβ² π = β π¬π’π§π π + ππ¨π¬π π πβ² π = β π¬π’π§π π πβ² =β π π π π π π + ππ¨π¬ π π π π + π π 1 π π π =β + =β π π π 15 EXAMPLE 9: Differentiate π π₯ = sin 3π₯ cos π₯ 2 Using the Quotient Rule and Chain Rule π π ππ¨π¬ ππ β π¬π’π§ ππ β π¬π’π§ ππ (ππ) ππ¨π¬ π πβ² π = ππ¨π¬ ππ π π )(ππ¨π¬ ππ) + ππ(π¬π’π§ ππ) π¬π’π§ ππ π(ππ¨π¬ π πβ² π = ππ¨π¬ ππ π 16 2 2 EXAMPLE 10: Differentiate f ο¨ x ο© ο½ 3x cos ο¨ 2 x ο© Using the Product and Chain Rules. g ο¨ x ο© ο½ 3x 2 g ' ο¨ x ο© ο½ 6x h ο¨ x ο© ο½ cos ο¨ 2 x ο© ο½ ο¨ cos 2 x ο© 2 2 h ' ο¨ x ο© ο½ 2cos ο¨ 2 x ο© ο¨ ο sin ο¨ 2 x ο© ο© ο¨ 2 ο© h ' ο¨ x ο© ο½ ο4 ο¨ cos ο¨ 2 x ο© sin ο¨ 2 x ο© ο© f ' ο¨ x ο© ο½ cos 2 (2 x)[6 x] ο« 3x 2 ο©ο« ο4 ο¨ cos ο¨ 2 x ο© sin ο¨ 2 x ο© ο©οΉο» f ' ο¨ x ο© ο½ 6 x cos 2 (2 x) ο 12 x 2 cos ο¨ 2 x ο© sin ο¨ 2 x ο© 17 2 2 EXAMPLE 10: Differentiate f ο¨ x ο© ο½ 3x cos ο¨ 2 x ο© Using the Product and Chain Rules. g ο¨ x ο© ο½ 3x 2 g ' ο¨ x ο© ο½ 6x Alternate approach h ο¨ x ο© ο½ cos ο¨ 2 x ο© ο½ ο¨ cos 2 x ο© 2 2 h ' ο¨ x ο© ο½ 2cos ο¨ 2 x ο© ο¨ ο sin ο¨ 2 x ο© ο© ο¨ 2 ο© h ' ο¨ x ο© ο½ ο4 ο¨ cos ο¨ 2 x ο© sin ο¨ 2 x ο© ο© Using the sine double angle identity sin 2A = 2sin A cos A h ' ο¨ x ο© ο½ ο2 ο¨ 2cos ο¨ 2 x ο© sin ο¨ 2 x ο© ο© ο½ ο2sin ο¨ 4 x ο© f ' ο¨ x ο© ο½ 3x 2 ο¨ ο2sin 4 x ο© ο« cos2 ο¨ 2 x ο©ο¨ 6 x ο© f ' ο¨ x ο© ο½ ο6 x2 sin ο¨ 4 x ο© ο« 6 x cos 2 ο¨ 2 x ο© 18 Assignment Questions Differentiate each of the following trigonometric functions. 1 a) y = 4cos x dy ο½ ο4sin x dx 1 b) f(x) = 3sin(5x2) f ' ο¨ x ο© ο½ 3cos ο¨ 5x 2 ο© ο¨10 x ο© f ' ο¨ x ο© ο½ 30 x cos ο¨ 5x 2 ο© sin 2 x 1 c) y ο½ cosο¨2 x ο© 2 cos 2 x 2sin x cos x ο sin x ο¨ ο2sin 2 x ο© ο¨ ο© dy ο½ dx cos 2 ο¨ 2 x ο© 19 Assignment Questions Differentiate each of the following trigonometric functions. d) f(x) = β 2cos3(7x) ο½ ο2 ο¨ cos ο¨ 7x ο© ο© 3 REWRITE f ' ο¨ x ο© ο½ ο6 ο¨ cos ο¨ 7 x ο© ο© ο¨ ο sin ο¨ 7 x ο©ο¨ 7 ο© ο© 2 f ' ο¨ x ο© ο½ 42cos2 ο¨ 7 x ο© sin ο¨ 7 x ο© e) y = x sin x y ' ο½ x ο¨ cos x ο© ο« sin x ο¨1ο© y ' ο½ x cos x ο« sin x 20 Assignment Questions 2. Find the slope of the tangent to the function y = cos2(2x) at the point xο½ ο° 2 . dy ο½ 2 ο¨ cos ο¨ 2 x ο© ο© ο¨ 2 ο© ο¨ ο sin ο¨ 2 x ο© ο© dx dy ο½ ο4cos ο¨ 2 x ο© sin ο¨ 2 x ο© dx The slope of the tangent at xο½ dy ο½ ο2 ο¨ 2cos ο¨ 2 x ο© sin ο¨ 2 x ο© ο© dx dy ο½ ο2 ο¨ 2sin ο¨ 2 x ο© cos ο¨ 2 x ο© ο© ο½ ο2sin ο¨ 4 x ο© dx dy ο¦ ο¦ ο° οΆοΆ ο½ ο2sin ο§ 4 ο§ ο· ο· ο½ ο2sin ο¨ 2ο° ο© ο½ ο2 ο¨ 0 ο© ο½ 0 dx ο¨ ο¨ 2 οΈοΈ ο° 2 is 0 21