Lec 6 Differentiation Rules PDF

# LECTURE NO. (6) ## **Basic Rules of Differentiation** ### **Derivatives of Trigonometric Functions** **Example (5):** Find the first derivative for the following functions 1. y = cos(5x + 4)³ dy/dx = (-sin(5x + 4)³) (3(5x + 4)² × 5) = -15(5x + 4)² sin(5x + 4)³ 2. y = cos(2x² + 3)...

# LECTURE NO. (6) ## **Basic Rules of Differentiation** ### **Derivatives of Trigonometric Functions** **Example (5):** Find the first derivative for the following functions 1. y = cos(5x + 4)³ dy/dx = (-sin(5x + 4)³) (3(5x + 4)² × 5) = -15(5x + 4)² sin(5x + 4)³ 2. y = cos(2x² + 3) dy/dx = (4 cos³ (2x² + 3))(-sin(2x² + 3))(4x) 3. y = cos√x dy/dx = 1 / (2√x) sin√x 4. y = cos⁵x dy/dx = -5 cos⁴x sin x 5. y = cos(ax + b) dy/dx = -a sin(ax + b) **Example (6):** Find the first derivative for the following functions * y = tan(x²) → y' = 6. tan⁵(x²).2x * y = (csc x + ln x)⁶ → y' = 6. (csc x + ln x)⁵. (-csc x. cot x + 1/x) * y = sin(lnx) + sec(2x + 3) → y' = cos(lnx)/x + 2sec(2x + 3). tan(2x + 3) * y = (4⁴ + cos[x³])⁻⁴ → y' = -4(4⁴ + cos[x³])⁻⁵. (4⁴. ln 4 + sin x³. 3x²) ### **Example** Differentiate the functions: 1. y = x³ + sin x y' = 3x² + cos x 2. y = √x cosh x y' = √x (sinh x) + cosh x (1 / 2√x) 3. y = (√x³ + cos x) / sin x y' = ((3/2)x¹/² - sin x) sin x - (cos x)(x³/² + cos x) / sin² x ### **Differentiate the functions** 1. y = sin(x²) y' = cos(x²) × (2x) 2. y = cos(√x³ + sin x) y' = -sin(√x³ + sin x) × 1 / (2√x³ + sin x) × (3x² + cos x) ### **Examples** y = √(tan(1/cosh x)) = [tan(sech x)]¹/⁵ y' = 1/5 [tan(sech x)]⁻⁴/⁵ × sec²(sech x) × -sech x tanh x *** ### **(2) y = eˣ³ (ln(cot² x))** y' = eˣ³ × [ 1 / (cot² x) × (2 cot x) × (-csc² x) ] + (ln(cot² x) ) × [eˣ³ × 3x²] *** # THE END