1. y' = 7(x² + 4x + 6)⁶(2x + 4)
2. f'(x) = 5(x³ - 5x)⁴(3x² - 5)
3. f'(x) = 10(3x - 2)⁹(5x² + 1)¹²(3) + (3x - 2)¹⁰·12(5x² + 1)¹¹(10x)
4. f'(x) = 3(6x² + 5)²(12x)(x² - 7)⁴ + (6x² + 5)³·4(x² - 7)³(2x)
5. y' = -8(2x² - 6x + 1)⁻⁹(4x - 6)
6. y' = (1/2)(x² - 7x)⁻¹/²(2x - 7)
7. y' = -4(x² - 2x - 5)⁻⁵(2x - 2)
8. f'(x) = (3/2)(x - 1/x)¹/²(1 + 1/x²)
9. y' = [1·(x + 7) - (x - 6)·1]/(x + 7)² = 13/(x + 7)²
10. y' = (-1/3)(2x - 1)⁻⁴/³(2) = -2/(3(2x - 1)⁴/³)
11. y' = sec²(3x)·3 = 3sec²(3x)
12. y' = 4sec(5x)tan(5x)·5 = 20sec(5x)tan(5x)
13. y' = -sin(x²)·2x = -2x sin(x²)
14. y' = 3cos²(x)(-sin(x)) = -3cos²(x)sin(x)
15. f'(x) = 8(1 + cos²x)⁷·2cos(x)(-sin(x)) = -16cos(x)sin(x)(1 + cos²x)⁷
16. y' = sec²(x²)·2x + 2tan(x)sec²(x)
17. y' = -sin(tan x)·sec²(x)
18. y' = 2sin(x)cos(x) - 3cos²(x)sin(x)
19. y' = 2sin(cos(4x))·cos(cos(4x))·(-sin(4x))·4 = -8sin(4x)sin(cos(4x))cos(cos(4x))
20. y' = [2sin(x)cos(x)·cos(x) - sin²(x)(-sin(x))]/cos²(x) = [2sin(x)cos²(x) + sin³(x)]/cos²(x)
21. y' = 3sin²(2x + 3)·cos(2x + 3)·2 = 6sin²(2x + 3)cos(2x + 3)
Parent Tip: Review the logic above to help your child master the concept of chain rule for derivatives worksheet.
