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.

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

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