Answer for problem a y = [f(x)]ⁿ → dy/dx = n f(x)]ⁿ⁻¹ f'(x) dy/dx = cos² (x³) = 2 cos (x³) [-sin (x³)] (3x²) = -6x² sin (x³) cos (x³) Because 2 sin (A) cos (A) = sin (2A): dy/dx= -3 sin (2x³)
Answer for problem b y = 1 / (3x - 2) = (3x - 2)⁻¹ dy/dx = -(3x - 2)⁻² (3) = -3 / (3x - 2)²
Answer for problem c y = (5x³ - x⁴)⁷ dy/dx = 7(5x³ - x⁴)⁶ (15x² - 4x³) = 7 [x³(5 - x)]⁶ x²(15 - 4x) = 7x¹⁸(5x - x)⁶ x²(15 - 4x) = -7x²⁰ (x - 5)⁶ (4x - 15)
Verified answer
Calculus Derivative.Answer for problem a
y = [f(x)]ⁿ → dy/dx = n f(x)]ⁿ⁻¹ f'(x)
dy/dx = cos² (x³)
= 2 cos (x³) [-sin (x³)] (3x²)
= -6x² sin (x³) cos (x³)
Because 2 sin (A) cos (A) = sin (2A):
dy/dx= -3 sin (2x³)
Answer for problem b
y = 1 / (3x - 2) = (3x - 2)⁻¹
dy/dx = -(3x - 2)⁻² (3)
= -3 / (3x - 2)²
Answer for problem c
y = (5x³ - x⁴)⁷
dy/dx = 7(5x³ - x⁴)⁶ (15x² - 4x³)
= 7 [x³(5 - x)]⁶ x²(15 - 4x)
= 7x¹⁸(5x - x)⁶ x²(15 - 4x)
= -7x²⁰ (x - 5)⁶ (4x - 15)