Unit 7 - Objective 1 - Derivatives of the Sine and Cosine Functions

All the rules for derivatives (products, quotients, chain rule, and implicit differentiation) still apply for trigonometric functions.

Derivative for sin(u)

Problem 1: Find the derivative of

y = sin 7x	u = 7x = angle
= cos 7x (7x) = (cos 7x) (7) = 7 cos 7x	(7 is written in front of the trigonometric function to separate it from the angle.)

Problem 2: Find the derivative of

y = 3x² sin²x	[this is a product (3x²) (sin²x) = uv]
Remember sin²x = (sin x)² = u²

Derivative for cos(u)

Problem 1: Find the derivative

y = 7 sin x cos 8x	(you need the product rule)
= 7 sin x (cos 8x) + cos 8x (7 sin x)
= 7 sin x (-sin 8x) (8x) + (cox 8x) (7 cos x)
= 7 sin x (-sin 8x) (8) + (cox 8x) (7 cos x)
clean it up
= -56 sin x sin 8x + 7 cos 8x cos x

Problem 2: Find the derivative

Problem 3: Find the derivative

2x cos y + sin 4x = 5	(use implicit differentiation)
(2x cos y + sin 4x) = (5)
You need the product rule for the first term.
2x (cos y) + cos y (2x) + sin 4x = (5)
2x (-sin y) cos y (2) + cos 4x (4x) = 0
2x (-sin y) cos y (2) + (cos 4x) (4) = 0
Clean it up
-2x sin y 2 cos y + 4 cos 4x = 0
You need by itself.
-2x sin y = -2 cos y - 4 cos 4x